Linear Transformations
T/F: If the linear transformation T(x)=Ax maps Rn to Rn, then A has n pivot positions.
False, we don't know enough about the transformation.
T/F: The columns of an invertible nxn matrix form a basis for Rn.
True, an invertible nxn matrix has n linearly independent columns, which can form a basis for Rn.
T/F: If v is a vector in Col(A), then Av=0.
False, this is the definition for Nul(A). A vector is in Col(A) if Ax=v has at least one solution x.
T/F: A square matrix A is invertible if det(A) = 0
False.
T/F: Vectors u and v are orthogonal if u⋅v = 0.
True
T/F: If A is an nxn matrix and the transformation T(x)=Ax is one-to-one, then the transformation is also onto.
True, the transformation is also onto by the Invertible Matrix Theorem.
T/F: If dim(V) = p, then there exists a spanning set of p+1 vectors in V.
True, the additional vector can be linearly dependent to any vector in the basis, or we could even use the zero vector.
T/F: Let U be a subspace of R5. dim(U) ≤ 5.
True, a basis for R5 has at most 5 vectors.
T/F: If A is diagonalizable, then A is invertible.
False. If 0 is an eigenvalue of A, then Ax=0 has a nontrivial solution.
T/F: If U is an orthogonal matrix, then it is invertible and U-1 = UT.
True
T(x,y) = (2x-3y, x+4, 5y)
Is T a linear transformation? Justify your response.
No, T(0)≠ 0.
If a 3x5 matrix A has rank(A) = 3, find dimNul(A) and rank(AT).
dimNul(A) = 2,
rank(AT) = 3
Is B = {(1, 0, 0), (1, 1, 0), (1, 1, 1)} a basis for R3?
Yes, B is a linearly independent spanning set of 3 vectors in R3.
A is a 3x3 matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?
No, we need to have 3 linearly independent eigenvectors.
Find the distance between vectors x = (10,-3) and y = (-1, -5).
dist(x,y) = sqrt(125) = 5*sqrt(5)
T(a,b,c) = (a-5b+4c, b-6c)
Find the standard matrix of T. Is T one-to-one, onto, both, or neither?
A =
[1 -5 4;
0 1 -6]
T is onto.
A = [3 0 2;
6 0 4;
0 0 0]
Is v = (-2, 2, 3) in Nul(A)?
Yes, Av = 0.
Let A = [-6 12;
-3 6 ]
and w = (2, 1).
Is w in Col(A)?
Yes, Ax = w is consistent
Find the area of the parallelogram whose vertices are (-2,0), (0,3), (1,3), (-1,0).
Area = 3.
Is the set {(3,-1,2), (1,-1,-2)} orthogonal? Is it orthonormal?
It is orthogonal, but not orthonormal.
Let T: Rn → Rm. Find an inequality for m and n when 1) T is onto 2) T is one-to-one.
1) onto: m ≤ n
2) one-to-one: m ≥ n
W = {(4a+3b, 0, a+b+c, c-2a) : a,b,c are real numbers}
Is W a subspace? Justify your reasoning.
W is a subspace.
W = span{(4, 0, 1, -2), (3, 0, 1, 0), (0, 0, 1, 1)}
H = {(x,y,z) : 2x+y=z, z=-x, x=3y-z}. Is H a subspace of R3? Justify your reasoning.
H is a subspace. H = Nul(A) where A =
[2 1 -1;
1 0 1;
1 -3 1;]
Let A = PDP-1. Find A4.
P = [5 7; and D = [2 0;
2 3] 0 1]
A4 = [226 -525;
90 -209]
y = (1, 3, 5), u = (1, 3, -2), v= (5, 1, 4)
Find the orthogonal projection of y onto Span{u,v} and its orthogonal complement.
projection = (10/3, 2/3, 8/3) and complement = (-7/3, 7/3, 7/3).