A subset W of Rn is a subspace of Rn if and only if the following conditions are met: vectors in any basis for the null space is a subspace of R3 1. The zero vector is in W. 2. x + y is in W whenever x and y are in W 3. ax is in W whenever x is in W and a is any scalar number of basis vectors

1. The zero vector is in W. 2. x + y is in W whenever x and y are in W 3. ax is in W whenever x is in W and a is any scalar

If S={v1,..., vr} is a set of vectors in Rn, then the set W consisting of all linear combinations of v1,..., vr is 1. The set is linearly dependent 2. The set does not span Rn n a subspace of Rn and is called the subspace spanned by S If A is an (m × n) matrix, then n = rank(A) + nullity(A)

a subspace of Rn and is called the subspace spanned by S

Let A be an (m × n) matrix. The range of A [denoted R(A)] is the set of vectors in Rm defined by the subspace spanned by the column vectors of the matrix NO R(A) = {y: y = Ax for some x in Rn}. exactly n linearly independent vectors

R(A) = {y: y = Ax for some x in Rn}.

the range of a matrix is the same thing as the column space of that matrix. The column space is maximum number of linearly independent vectors that can exist in the space the subspace spanned by the column vectors of the matrix so that there isnt a row of the form [0 0 0 0 1] have the same dimension.

the subspace spanned by the column vectors of the matrix

the span of a set of vectors is the same thing as the range of a matrix whose columns are those vectors YES basis If A is an (m × n) matrix, then n = rank(A) + nullity(A)

the range of a matrix whose columns are those vectors

the dimension of a vector space is the basis the rank of AT NO maximum number of linearly independent vectors that can exist in the space

maximum number of linearly independent vectors that can exist in the space

Ace Your Linear Algebra Exam

Flashcard

Learn Mode

Match

Library

Create

Flashcards

Library

Match (Coming Soon)

Algebra - Linear Algebra

user_hodr Created by 9 mon ago

Cards in this deck(32)

A subset W of Rn is a subspace of Rn if and only if the following conditions are met:

If S={v1,..., vr} is a set of vectors in Rn, then the set W consisting of all linear combinations of v1,..., vr is

any plane through the origin in R3

Let A be an (m × n) matrix. The null space of A [denoted N (A)] is the set of vectors in Rn defined by

Let A be an (m × n) matrix. The range of A [denoted R(A)] is the set of vectors in Rm defined by

the range of a matrix is the same thing as the column space of that matrix. The column space is

In words, the null space consists of all those vectors x such that Ax

the span of a set of vectors is the same thing as

If you have a set of vectors where the number of vectors exceeds the dimension of the space, the vectors in the set will always be

the dimension of a vector space is the

a linearly independent spanning set is also called a

the dimension of a subspace is the

does every basis have the same number of vectors?

are bases unique?

Let W be a subspace of Rn, and let B = {w1, w2,..., wp} be a spanning set for W containing p vectors. Then any set of p +1 or more vectors in W is

Let W be a subspace of Rn with dim(W ) = p. 1. Any set of ______ or more vectors in W is linearly _________. 2. Any set of fewer than _____ vectors in W does not ______ W. 3. Any set of ______ linearly ___________ vectors in W is a ______ for W. 4. Any set of _____ vectors that _______ W is a basis for W

what is the dimension of the null space called?

what is the dimension of the range called?

the rank is the number of

the nullity is the number of

If A is an (m × n) matrix, then the rank of A is equal to

If A is an (m × n) matrix, then the row space and the column space of A

An (m × n) system of linear equations, Ax = b, is consistent if and only if

why do the ranks of A and [A | b] have to be equal?

the number of leading ones

An (n × n) matrix A is nonsingular if and only if the rank of A

the nxn identity matrix has rank

for a nonsingular matrix, the null space includes only

linearly independent solutions to Ax = 0 are always

what does the rank-nullity theorem state?

a set of vectors does not form a basis for Rn if

in order for a set of vectors to form a basis for Rn, there must be how many vectors in the set and how must they be related?

Ask Our AI Tutor

Get Instant Help with Your Questions

Need help understanding a concept or solving a problem? Type your question below, and our AI tutor will provide a personalized answer in real-time!

How it works

Ask any academic question, and our AI tutor will respond instantly with explanations, solutions, or examples.

Get Started

Browse questions and discover topic-based flashcards
Practice with engaging flashcards designed for each subject
Strengthen memory with concise, effective learning tools

Discover By Topic

Ace Your Linear Algebra Exam

Related Decks