Applications of determinants

Be able to use determinants to:

• find areas of parallelograms and volumes of parallelepipeds

• solve systems of equations by Cramér’s Rule

• find entries in the inverse of a matrix

Vector spaces, subspaces, linear independence and bases

Be able to:

• recognize when a subset of a vector space is a subspace using the Subspace Test, or the fact that the following are ALWAYS subspaces: Span{v₁, ... v_p}, Col(A), Nul(A).

• determine whether a given set of vectors in a general vector space is linearly independent or linearly dependent (with reasons)

• determine whether a set of vectors is a basis for Rⁿ

• state the standard basis for Rⁿ, for P_n, for M_m_{x n}

• find the coordinates of a vector with respect to a basis

• find a vector, given its coordinates

• use coordinates to test for linear independence

• check whether or not a given set of vectors is a basis

• reduce a spanning set of a subspace H to a basis for H

Dimension

Be able to:

• find the dimension of instances of Rⁿ, of P_n, of M_m_{x n}

• find the dimension of, and basis for, Col A, Nul A and Row A

• use the fact that dim Col A + dim Nul A = number of columns of A

• use the fact that rank(A) = dim Col(A) = dim Row(A)

Applications

Be able to:

• find a steady-state probability vector for a stochastic matrix, and interpret its meaning in the context of an application

• answer questions about linear systems that use the concepts mentioned on this page

• answer true/false questions about the concepts mentioned on this page