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82 Cards in this Set
- Front
- Back
row equivalence
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sequence of simple row operations that make one matrix into other
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row equivalence and augmented matrices
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if two augmented matrices are row equivalent, then two systems have same solution set
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what makes something in echelon form?
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zeros are at the bottom of the matrix, not at the top
each leading entry of a row is to the right of the one above it all entries in a column below a leading entry are zeros |
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RREF classification?
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leading entry must be 1
leading entry is the only nonzero number in that column |
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what is a leading entry?
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leftmost nonzero entry
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pivot position properties vs. pivot properties
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in RREF, pivot positions are fixed no matter what multi or add you do
pivot is the actual number in the column, can be many different numbers |
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basic variables
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variables corresponding to pivot columns
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general solution/parametric descriptions form
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x1=gsdgs
x2=3sgfg x3 is free thus, it gives explicit descriptions of all solutions |
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parametric description
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whenever a solution set has free variables/consistent, it has many parametric descriptions
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how to write basic variables in general solution?
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write basic variables in terms of free variables
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vector
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a list of numbers, a one column matrix
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[w1]
[w2] = w what are w1 and w2? |
ordered pairs
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vectors = vectors iff
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corresponding entries are equal
7 4 4 does not equal 7 |
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R^n
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n= how many rows
R is real numbers that appear as entries in the vectors |
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R^2
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2x1 column matric
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R^3
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3x1 column matrix
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zero vector
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a vector whose entries are all zero, represent one single point
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q: determine whether b is a linear combo of a1 a2
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x1a1 +x2a2 = b
so row reduce the augmented matrix to see if it is consistent. if consistent, than b is a linear combo of a1a2 |
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the span of {v1...vp} is the set
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of all vectors linear combinations
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q: is b is in the span{v1..vp}
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is b a linear combination??
thus x1v1 +x2v2 = b? |
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geometrically defining span
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the line/plane with all the possible linear combinations
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zero vector & span
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0 vector must be always be in span
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AX=B
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A is an m*n matrix with all the vectors
X representes weights B is the product of A and X, assuming x is in R^n |
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vector equation
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x1a1 +x2a2 +x3a3 =b
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matrix equation
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[a1 a2 a3][x1]
[x2] [x3] |
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statements that are logically equivalent
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for each b in R^m the equation AX=B has a solution
each B in R^m is a linear combo of colums of A the columns of A span R^m A has a pivot position in every row |
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homogenous system
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AX=0
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trivial solution
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when x=0 is a solution
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non trivial solution
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for AX=0, nonzero vector x that satisfies the equation
AX=0 will only have a nontrivial solution if and only if the equation has at least one free variable |
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how to write solution set for homogeneous systems
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x=[], then pull out x1 or x3 so you have
xv |
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how to describe solution set for a homogeneous equation
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span {v1..vp}
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how to write a solution to AX=B in parametric vector form
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x2[] +x3[] = x
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translation
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vector additon
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definition of translation geometrically
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moves vector x in a parallel direction by adding p
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if a set contains a zero vector
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linearly dependent
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linear independence
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no free variables
columns of matrix A are linearly independent if Ax=0 has only the trivial solution SETS OF ONE/TWO VECTORS: if a set containing one vector is not a zero vector SETS OF TWO OR MORE VECTORS: rows>columns don't mean shit |
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linearly dependence
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free variables that don't give solutions that are all zeros
an indexed set S={v1...vp} is linearly dependent iff at least one vector in S is a linear combo of others SETS OF ONE OR TWO VECTORS if v1 is a multiple of v2, sets are linearly dependent if a set contains a 0 vector SETS OF TWO OR MORE VECTORS: if columns > rows (more variables than equations, so there must be a free variable) |
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T:R^n to R^m
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transformation, function, mapping
assigns to each vector x in R^n a vector t(x) in R^m |
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R^n in transformations
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domain of T
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R^m in transformations
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codomain
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image of x
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for x in R^n, the vector T(x) in R^m
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range
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the set of all images T(x) is called range
set off all linear combinations of columns of A because T(x) = Ax |
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matrix transformations
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T(x) behaves like Ax
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Find T(u) the image of u under the transformation
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Compute T(u) which is Au
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Find an x in R^2 whose image under T is b
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Solve T(x)=b for x, or solve Ax= b
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Is there more than one x whose image under T is b?
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Is the system unique or not unique?
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Determine if c is in the range of the trans T
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Is T(x)=c consistent?
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shear transformation
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produces parallelogram
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A transformation is linear if
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T(u+v) = Tu + Tv
T(cu) = cT(u) |
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matrix transformations =
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linear transformation
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horizontal contraction/dialation
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k is in upper left
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vertical contraction/dialation
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k is in lower right (diagonal to horizontal contraction)
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horizontal shear
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k is in the upper right hand
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vertical shear
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k is in the lower left hand
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onto
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T maps if and only if columns
existence question span R^m or every vector in R^m is line no zero row in REF of A |
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one to one
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uniqueness question
pivot in each column has a unique solution or none at all |
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not onto
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existence question
when there is no solution |
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not one to one
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uniqueness question
more than one solution |
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scalar multiple
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rA is a scalar multiple of A when r is a scalar
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commute
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if AB = BA
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AB= AC, B does not equal C
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True
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transpose of A is:
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n*m matrix when A is m*n
denoted by ^T |
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(A^T)^T =
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A
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(AB)^T
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B^T * A^T
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invertible matrix A and C theorem
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both must be square and when you multiply them together, it equals I
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singular matrix
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matrix that is not invertible
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nonsingular matrix
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invertible matrix
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ad-bc is...
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determinant
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test for invertibility
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ad-bc != 0
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If A is an invertible n*n matrix for each b in R^n
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the equation Ax=b has a unique solution
x= (A^-1)b A((A^-1)b=b b=b (true!) |
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invertible matrix is row equivalent to an identity matrix how?
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by watching the row reduction of A to I
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a square matrix A is invertible iff A is row equivalent to an identity matrix or after you perform elementary row operations that reduce A to In
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memorize statement
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IMT
A is row |
equivalent to the n*n identity matrix
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IMT
A has |
n pivot positions
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IMT
The equation Ax=0 |
has only the trivial solution
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IMT
The columns |
of A form a linearly independent set
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IMT
The linear transformation x |--> Ax |
is one to one
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IMT
The equation Ax=b |
has at least one solution for each b in R^n
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IMT
The columns |
of A span R^n
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IMT
The linear transformation x |--> Ax |
maps R^n onto R^n
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A^T is
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an invertible matrix
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There is an n*n matrix D such that
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AD=I
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