Test 2 Flashcards
If f is a function in the vector space V of all real-valued functions of R and if f(t)=0 for some t, then f is the zero vector in V.
FALSE. while f(t) = 0 for some t might be a property of a function, it doesn’t guarantee that the function is the zero vector in the vector space
The Null Space of A is the solution set of the equation Ax = 0
TRUE
Col A is the set of all SOLUTIONS to Ax = b
FALSE - Col(A) determine all the possible OUTPUT, not solution
If the equation Ax = b is consistent, then Col(A) = R^m
FALSE - Col(A) might not Span all of R^m.
The RANGE of a linear transformation is a vector space
TRUE - also the range is a subspace of the output vector space W.
A single vector by itself is linear dependent
FALSE - single vector by itself cannot be expressed as a linear combination of any other vectors, including itself (with a zero scalar).
If H = Span{b1…bp}, then {b1…bp} is a basis for H
TRUE
If V = R^n and C is the std basis for V, then P(C<-B) is the same as the change of coordinates matrix PB.
TRUE
If v1…vp are in R^n , then Span{v1…vp} is the same as the column space of the matrix [v1…vp]
TRUE
If B ={v1…vp} is a basis for a subspace H and if x =c1v1 +…+cpvp, then c1,…cp are the coordinates of x relative to the basis B.
TRUE
The dimension of Col A is the number of pivot columns of A.
TRUE
Each line in R^n is a one-dimensional subspace of R^n
FALSE - could be 0
If P[B] is the change-of-coordinates matrix, then [x]B = P[B].x, for x in V
FALSE - b/c [x]B = x . (P[B])^-1
A plane in R^3 is a two-dimensional subspace of R^3
TRUE
The dimension of the vector space of signals S is 10
FALSE
Given A an n x n invertible matrix, the following are equivalent
- Columns of A form a basis of R^n
- Col A = R^n
- Rank A = n
- Nullity A = 0
- Nul(A) = {0}
If V = R^n and C is the std basis for V, then P(C<-B) is the same as the change of coordinates matrix P(B) in 4.4
TRUE
An n x n determinant is defined by determinants of (n-1) x (n-1) submatrices.
FALSE - missing cofactor
a row replacement operation does not affect the determinant of a matrix
TRUE - row replacement det A = det B
If the columns of A are linearly dependent , then det A = 0
TRUE - colinear lines will have zero area (det = 0)
If three row interchanges are made in succession, then the new determinant equals the old determinant
FALSE - each row interchanges multiply det by (-1); three times will made det A = - det B
The set of 2x2 matrices with positive determinant is a subspace
FALSE - not enough information to satisfy conditions:
1. Closure under addition det(A+B) > 0
2. A + B > 0
The set of 2x2 matrices with zero determinant is a subspace
FALSE - det(A+B) might not = 0 and also linear dependent row/col might not sustain once added
If A is a 2x2 matrix with a zero determinant, then one column of A is a multiple of the other
TRUE - det = 0 means linear dependent row/col
If B is produced by multiplying row 3 of A by 5, then det B = 5. det A
TRUE
det ( A^T . A ) >= 0
TRUE when A is positive definite matrices
The set of all linear combinations of v1,…,vp is a vector space
TRUE
If S is linearly independent, then S is a basis for V
FALSE - only one condition satisfied, S also need to span space V
If Span S = V, then some subset of S is a basis for V
TRUE
Determine if w = [. . .] is in subspace of R^3 created by v1=[. . .] and v2=[. . .]
w must be in Span (v1,v2) or w must be the result of a linear combination of v1,v2 ( c1v1 + c2v2 = w)
Determine given sets { [. .]; [. .] } that can become basis for R^2 or R^3
- Check det; if det is non-zero -> invertible -> sets are basis for R^2 / R^3
- Must have n pivot position in n row for R^n
Rank of matrix A is dimension of pivot column space A
TRUE - n pivot columns -> rank = n
Rank(A) + dim Nul(A) = n
TRUE - n = total number of columns in matrix A
Find coordinate of x in subspace B generated by basis b1,b2
by finding scalars c1.b1 + c2.b2 = [x1; x2]
[x]B = [x1; x2]
