Math 18 Final Review

Linear Transformations

100

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.

200

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.

300

T(x,y) = (2x-3y, x+4, 5y)

Is T a linear transformation? Justify your response.

No, T(0)≠ 0.

400

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.

500

Let T: Rⁿ → R^m. 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

Vector Spaces 1

100

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.

200

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.

300

If a 3x5 matrix A has rank(A) = 3, find dimNul(A) and rank(A^T).

dimNul(A) = 2, 

rank(A^T) = 3

400

A = [3 0 2;

        6 0 4;

        0 0 0]

Is v = (-2, 2, 3) in Nul(A)?

Yes, Av = 0.

500

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)}

Vector Spaces 2

100

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.

200

T/F: Let U be a subspace of R⁵. dim(U) ≤  5.

True, a basis for R⁵ has at most 5 vectors.

300

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.

400

Let A = [-6 12;

             -3 6 ]

and w = (2, 1).

Is w in Col(A)?

Yes, Ax = w is consistent

500

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;]

Diagonalization

100

T/F: A square matrix A is invertible if det(A) = 0

False.

200

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.

300

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.

400

Find the area of the parallelogram whose vertices are (-2,0), (0,3), (1,3), (-1,0).

Area = 3. 

500

Let A = PDP^-1. Find A⁴.

P = [5 7; and D = [2 0;

       2 3]               0 1]

A⁴ = [226 -525;

          90  -209]

Orthogonality

100

T/F: Vectors u and v are orthogonal if u⋅v = 0.

True

200

T/F: If U is an orthogonal matrix, then it is invertible and U^-1 = U^T.

True

300

Find the distance between vectors x = (10,-3) and y = (-1, -5).

dist(x,y) = sqrt(125) = 5*sqrt(5)

400

Is the set {(3,-1,2), (1,-1,-2)} orthogonal? Is it orthonormal?

It is orthogonal, but not orthonormal.

500

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).