LA 4.3-4.4

linearly independent

An indexed set of vectors {v₁, … , v_p} in V if the vector eq. c₁v₁+c₂v₂+ … + c_pv_p = 0 has only the trivial solution, where c₁, … , c_p all = 0

linearly dependent

An indexed set of vectors {v₁, … , v_p} in V if the vector eq. c₁v₁+c₂v₂+ … + c_pv_p = 0 has a nontrivial solution, where some weights c₁, … , c_p are not all zero & the eq. holds

linear dependence relations among v₁, … , v_p

c₁v₁+c₂v₂+ … + c_pv_p = 0

c₁v₁+c₂v₂+ … + c_pv_p = 0

linear dependence relations among v₁, … , v_p

v ≠ 0

A set containing a single vector v is lin. ind. if

linearly independent

A set containing a single vector v is ______ if v ≠ 0

linearly dependent because the zero vector doesn’t contribute to the rest of the set

Any set of vectors containing a zero vector is

one of the vectors is a multiple of the other

A set of 2 vectors is lin. dep. if

linearly dependent

A set of 2 vectors is ________ if one of the vectors is a multiple of the other

some v_j (where j > 1) is a linear combination of the preceding vectors, v₁, … , v_j-1

An indexed set {v₁, … , v_p} of 2 or more vectors with v₁ ≠ 0 is lin. dep. iff

linearly dependent

An indexed set {v₁, … , v_p} of 2 or more vectors with v₁ ≠ 0 is ________ iff some v_j (where j > 1) is a linear combination of the preceding vectors, v₁, … , v_j-1

basis for a subspace H of a vector space V

a set of vectors β in V if

1. β is a lin. ind. set

2. the subspace spanned by B coincides w/ H. That is, H = Span B

1. B is a lin. ind. set

2. the subspace spanned by B coincides w/ H. That is, H = Span B

β is a basis if

Spanning Set Theory

A basis can be constructed from a spanning set by discarding unneeded vectors

A basis can be constructed from a spanning set by discarding unneeded vectors

Spanning Set Theory

H contains at least one nonzero vector

If subspace H ≠ 0, then

1. If one of the vectors, v_k, in S is a linear combination of the remaining vectors in S, then the set formed from S by removing v_k still spans H

2. If H ≠ 0, some subset of S is a basis for H

If S = {v₁, … , v_p} is a set in a vector V & H = Span {v₁, … , v_p}, the following are true

finding x in Ax= 0, which always produces a lin. ind. set when Nul A contains nonzero vectors

basis of Nul A can be found by

the same solution

If matrix A is row reduced to a matrix B, then both matrices have

finding the pivot columns of matrix A before any row reduction

basis of Col A can be found by

their row spaces are the same

If two matrices A & B are row equivalent, then

Finding the row echelon form of A, called matrix B, the basis is the nonzero rows of B

basis of Row A can be found by

deletion of vectors from a spanning set must stop when the set becomes lin. ind.

When the Spanning Set Theorem is used, the

standard basis for Rⁿ

the columns of the n x n identity I_n

The Unique Representation Theorem

If β = {b₁, … , b_n} is a basis for a vector space V, then for x in V, there’s a unique set of scalars, c₁, … , c_n s.t. x = c₁b₁+ … + c_nb_n

The Unique Representation Theorem equation

x = c₁b₁+ … + c_nb_n where b₁ , … , b_n are basis vectors and scalars c₁, … , c_n are unique to x

x = c₁b₁+ … + c_nb_n where b₁ , … , b_n are basis vectors and scalars c₁, … , c_n are unique to x

The Unique Representation Theorem equation

coordinates of x relative to the basis β

β-coordinates of x are the same as

β-coordinates of x

weights c₁ , … , c_n s.t. x = c₁b₁ + … + c_nb_n

[x]_B=
[c₁]
[ . ]
[ . ]
[ . ]
[c_n]

coordinate vector of (relative to β) or the β-coordinate vector of x

coordinate vector of (relative to β) or the β-coordinate vector of x

[x]_B=
[c₁]
[ . ]
[ . ]
[ . ]
[c_n]

coordinate mapping is determined by

mapping x | → [x]_B

mapping x | → [x]_B

coordinate mapping is determined by

the change of coordinates matrix from β to the standard basis in Rⁿ

P_B

vector eq. x = c₁b₁ + … + c_nb_n is equivalent to x = P_B[x]_B

If the basis β = {b₁, … ,b_n}, then the

x = P_B[x]_B

change-of-coordinate equation

P_B^-1x

[x]^B=

placing the basis vectors b₁, … , b_n as the columns of the matrix P_B = [b₁ b₂ …b_n]

P_{B is constructed by}

invertible by the Invertible Matrix Theorem

Since the columns of P_B form a basis for Rⁿ, P_B is

x |→ [x]_B is a one-to-one linear transformation from V to Rⁿ

If β = {b₁, … , b_n} is a basis for a vector space V, then the coordinate mapping

isomorphism from vector space V onto vector space W

a one-to-one linear transformation from V to W

the same

iso in Greek means

form or structure

morph in Greek means

[u]_B + [w]_B

[u + w ]_B =

r[u]_B

[ru]_B =