T/F for Lin ALG

False

A vector space consists for four entities: A set of vectors, a set of scalars and two operations. 

True

Every vector space V contains at least one subspace.

True

True

False

False

Let A, B be  m x n matrices. 

If  A is row-equivalent to B, then the row space of A is equal to the row space of B. 

True

True

True

True

Every vector space V contains at least one subspace.

True

1

False

True

Determine if it is possible to write v as a linear combination of u1, u2, u3.

v = (-1, 7, 2)

u1 = (1, 3, 5)

u2 = (2, -1, 3)

u3 = (-3, 2, -4)

False

The set of all fifth degree polynomials is a vector space. 

False

If W and U are subspaces of a vector space V, then the union of W and U is a subspace of V.

False

False

False

Two vectors in a vector space are linearly dependent if and only if one is a scalar multiple of the other. 

True

Row operations on a matrix A may change the dependency relationships among the columns of A.

True

The zero vector  v = (0, 0, 0) can be trivially written as a linear combination of the following vectors. That is, 0 = 0v1 + 0v2 + 0v3.  

Determine if it is also possible to write the zero vector as a linear combination of the following vectors in a nontrivial way. 

v1 = (1, 3, 2)

v2 = (1, 1, 3)

v3 = (2, 4, 5)

True

False

Every vector space contains exactly two subspaces.

False

False

True

False

True

True

If an m x n matrix B can be obtained from elementary row operations on an m x n matrix A, then the column space of B is equal to the column space of A.

False

The set of all real valued ordered pairs of the form (0, y) with standard operations form a vector space. 

True

True

True

False

S spans a line

False

If an m x n matrix B can be obtained from elementary row operations on an m x n matrix A, then the column space of B is equal to the column space of A.

False

 The matrix B is row-equivalent to the matrix A if there exists a finite
               number of elementary matrices E₁, E₂, . . . , E_k,  so that 
               B = E_kE_k-1 ^{. .  . .} E₁A.  

True → B is row-equivalent to A if and only if B can be obtained from A by a finite sequence of elementary row operations.    And each of these elementary row operations can be achieved by multiplying by an appropriate elementary matrix. 

  Every matrix has a multiplicative inverse. 

False → any n x n matrix A with a row of 0's will fail to have a multiplicative inverse. 

 If A is an n x n matrix and c is a nonzero scalar, then the determinant of the matrix cA is given by
 
                       det(cA) = c det(A). 

False → The correct formula would be:   det(cA) = cⁿdet(A)

The set of all fourth-degree polynomials with the standard operations is a vector space.

False →

False

True

Let A and B be n x n matrices.  If A is similar to B, then  A² is similar to B². 

True → Assume that A is similar to B.   So there is an invertible P such that 

               A = P^-1BP  

Square both sides to get A² = (P^-1BP)(P^-1BP) 

                                               = P^-1B(PP^-1)BP

                                                = P^-1BBP

                                                = P^-1B²P      

Thus, A² is similar to B².

True

The dimension of a linear transformation T from a vector space V to a vector space W is called the rank of T.

False

If |A| = |B|, then B is similar to A.

False

A set S of vectors in an inner product space is orthonormal if every vector is a unit vector and each pair of distinct vectors in S is orthogonal.

True

True

True

True

To apply the Gram-Schmidt process to a set of vectors, the vectors must be unit vectors.

False

An orthonormal basis created by the Gram Schmidt process does not depend on the order of the vectors in the basis.

False

False

False

True

False

True

The range of a linear transformation T: V → W is a subspace of W.

True

Orthogonal

All pairs dot products are 0

orthonormal

dot product is zero and a magnitude of 1 (||u||=1)

Linear transformations

scalar multiplication, summations of all.

Things that DONT apply (sin(t), rank(t), det(t))

True

Invertible

non-singular

square matris

det does not = 0

Ax = b has a unique solution

Non-Invertible

singular

det = 0

row or column of zeros

rank(T)

number of non zero rows in a matrix

Is t one to one?

if nullity or kern = 0

Is T onto?

only if its columns space the entire codomain eg. if tranformation dim(W) = 4 = rank(T) = 4

Vector Spaces won’t create when

there is an inequality or set limitis

Linear Indenpendence

det does not = 0

pivots

trivial solution Ax = b

Linear Dependence

det = 0

no pivots

row of zeros

non-trivial solution

If A is a nonsingular square matrix, then the system of linear equations
            Ax = b has a unique solution. 

True

    Multiplying a single row of a square matrix A by a nonzero constant results in the
              determinant of A being multiplied by the same nonzero constant.

True

False

isomorphic things

   If A and B are nonsingular matrices, then A + B is also nonsingular.

False-
As a counterexample, take any n x n matrix A, and let B = −A. 
Then A + B = O_nn, which is not invertible. 

 The transpose of the inverse of a nonsingular matrix is equal to the inverse of the transpose.

TRUE By Theorem 2.8 in the text, we have the formula: 

                   (A^T)⁻¹ = (A⁻¹)^T

If  A is row-equivalent to B, then the column space of A is equal to the column space of B.

False

Of all the possible scalar multiples of a vector v, the orthogonal projection of u onto v  is the vector closest to u.

True

Suppose that T₁ and T₂ are linear transformations from an n-dimensional vector space V into an m-dimensional vector space W, and that A₁ and A₂ are their respective standard matrices.

Then the standard matrix for the composition T = T_{2 °} T₁ is  A₂A₁.

True