Math 1152 - Test Two

Linear algebra (linear independence/dependence, spans, subsets, spanning sets, basis,

What is a spanning set? (intuitively)

A collection of vectors that can be used to create any other vector within a set.

What does a spanning set for R² consist of?

Two non-collinear vectors NOT INCLUDING the zero vector.

Does the zero vector have a basis?

No the zero vector doesn’t have a basis since it’s spanning set is the zero vector but the zero vector is not linearly independent.

T/F The spanning set for a vector space is unique?

False

Counter: in R² the spanning set can be any two non-collinear (non-zero vector) vectors.

How to check for linear independence

How to find the basis for a given set of vectors (Thrm BS)

1. Construct a matrix

2. REFF

3. Basis is formed by the vectors that correspond pivot columns

There exists a solution (x) to the linear system LS(A,b) IFF

b can be expressed as a linear combo of the columns in A.

Note: The entries of x are the scalar multiples for each column. (entry 1,1 is the scalar multiple of v_1 etc.)

T/F the set of n+1 vectors in R^n is linearly dependent?

True because the n+1 ensures there must be a free variable since there are more columns then equations (n<m) so there can’t be a pivot in every column.

T/F A set of n-1 vectors is always linearly independent?

False, because just because the matrix will have more equations than vectors (n<m) doesn’t ensure that the vectors are indepenent from one another.

T/F if each vector is not a scalar multiple of any other SINGULAR vector then the set is linearly independent.

False if each vector is not a scalar multiple of any other SINGULAR vector that doesn’t ensure that a vector is a linear combo of all the other vectors.

ex)

V dne aV2 or aV3, doest’t mean

V dne a2V2 +r a3V3

Intuitively what is the span of a set?

All possible linear combinations of that set.

T/F any vector in a vector space can be expressed as a linear combo of the vectors that make up the basis?

T/F If a set contains more than n vectors in R^n then the set is linearly dependent.

True

In other words if the number of vectors in a set exceeds the dimension of the space then its linearly dependent

What is a subspace?

A subset of a vector space. That is closed under scalar multiplication, addition and contains the zero vector.

What must a subset contain?

The zero vector.

Subspace vs Span vs Spanning set (intuitively)

A subspace is a subset of a vector space. (closed under addition, scalar multiplication and contains zero vector)

A span refers to all the possible linear combinations for a set of vectors.

A spanning set is a set of vectors whose span is equal to the entire vector space.

T/F if the REF of a matrix has a pivot in every column it is linearly independent.

True, since a pivot in every column frees to no free variables.

T/F a matrix with n>m can never be linearly independent.

True since there can never be a pivot in every column is there are more variables than equations.

What is a Basis

A linearly independent spanning set. 

Thrm BIS: Suppose V is a vector space with a finite basis B and a second finite basis C. Then:

B and C must have the same dimension.

thrm SSLT/fD: Suppose S={V1,V1…Vt} is a basis for the vector space V. Then any set with t+1 vectors is ________________

linearly dependent

T/F Suppose S={V1,V1…Vt} is a basis for the vector space V. Then any set with t-1 vectors is linearly independent?

False.

What is the standard basis for R^n?

(e1,e2,e3…..en}

What is the dim of R^n?

dim = n

Suppose dim V = t and S = {w1,….wm} is a set of vectors from V then what four thrms apply?

(hint: 1. m<t, 2.m>t, 3. m=t, 4.m=t)

1. if m<t then S is LD

2. if m>t then S does not span V. (for S to ‘span’ V it must span the ENTIRE space)

3. if m=t and S is LI then s spans V

4. if m=t and S spans V then S is LI

T/f If S is a linearly independent set in V, and w is a vector not in the span of S then adding w will keep the keep the set linearly independent.

True

Adding a vector to a set that is not in the spa of the set will keep the set linearly independent. (since if the vector you add isn’t  in the span of the set that also implies its not a linear combo of any other vector in the set by definition.

If U and W are both subspaces of V and W is a subset of U then:

if U DNE W then ___________________________

if dim U = dim W ___________________________

If U DNE W then dim U > dim W

if dim U = dim W: U=W

(If u and w are both subspaces and w is a subset of u either dim w<dim u or if dimu=dimw then they are identical)

T/F for matrix addition the matrices have to have the same dimensions?

True

Definition of matrix addition. (def MA)

[A^t] kj = _____

[A^t] jk

(rows and columns flip)

What is a symmetric matrix?

A matrix where B^t=B

(A matrix where the transpose is the same as the og)

What type of matrix CAN sometimes be symmetric?

A square matrix

How to take the conjugate of a complex matrix?

Just take the conjugate of each entry in the matrix.

Suppose that A and B are m×n matrices. Then  conjugate (A+B) = _______

conjugate (A+B) = conjugate(A) + conjugate(B)

conjugate (A*B) = _______

conjugate (A*B) = conjugate(A) * conjugate(B)

conjugte(A^t) = _______________

conjugte(A^t) = (conjugate(A))^t

What will be the dimension of the product matrix if you multiply an m×n by an n×p matrix?

an mxp matrix

(so the dimension of the product matrix is the rows of the first matrix and the columns of the 2nd)

What criteria is needed to multiply two matrices?

The columns of the first matrix must equal the rows of the 2nd?

Is matrix multiplication communitive?

No!

AxB DNE BxA

T/F is the RREF of a matrix has a pivot in every column it is a square matrix

False. The matrix can still have m>n

T/F if A is a square matrix and its RREF has a pivot in every column then its REFF is the identity matrix?

True

What is dim P2?

dim p² = 3

What is the standard basis for P2?

{1,x,x²}

A polynomial is equal to the zero polynomial iff

all it’s coefficients are equal to zero

(helpful for proving that a polynomial is linearly independent)

What is the definition of vector addition?