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Linear Algebra 2.9

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Section 1

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If a subspace H has a basis of p vectors,

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Algebra Math

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Mar 1, 2020

Cards (11)

Section 1

(11 cards)

If a subspace H has a basis of p vectors,

Front

then every basis of H must consist of exactly p vectors

Back

The Rank Theorem

Front

If a matrix A has n columns, then rank A + dimNulA = n.

Back

The Invertible Matrix Theorem (Continued)

Front

Let A be an nxn matrix. Then the following statements are each equivalent to the statement that A is an invertible matrix. 1) The columns of A form a basis of R^n 2) Col A = R^n 3) dimColA = n 4) rank A = n 5) NulA = {0} 6) dimNulA = 0

Back

The Basis Theorem

Front

Let H be a p-dimensional subspace of R^n. Any linearly independent set of exactly p elements in H is automatically a basis for H. Also, any set of p elements of H the spans H is automatically a basis for H.

Back

Dimension (of nonzero subspace H) (dimH)

Front

the number of vectors in any basis for H

Back

Coordinates of x relative to the basis B

Front

weights c1, ..., cp such that x=c1b1 + ... + cpbp, and the vector in R^p. *Suppose the set B={b1, ..., bp} is a basis for a subspace H

Back

Isomorphism (H is isomorphic to R^n)

Front

the correspondence x I-> [x]B is a one-to-one correspondence between H and R^n that preserves linear combinations (preserves the "look" of H)

Back

Coordinate Vector of x (Relative to B) (otherwise known as the B-Coordinate Vector of x)

Front

[x] = [ c1 ] B [ ... ] [ cp ]

Back

How do you determine the dimension of NulA?

Front

Count the number of free variables in Ax=0

Back

Rank (of Matrix A) (rank A)

Front

the dimension of the column space of A

Back

What is the dimension of the zero subspace?

Front

0

Back