linear test 1

The rank of a matrix equals the number of nonzero rows in its ROW ECELON FROM

the solution x=0 of a homogeneous equation Ax=0 (when the variables of a homogeneous equation/ matrix are equal to zero)

system where the constant term in each equation is zero (all solutions in the aug. matrix equal zero) if [A | 0 ] is the augmented matrix of a homogenous system & the system has more variables than equations, the system has infinitely many solutions

Any collection of vectors containing the zero vector must be linearly dependent.

The vectors are not scalar multiples of each other.

a set with no solution

A system of equations that has exactly one solution or an infinite # of solutions

a matrix that has an inverse

uu+2uv+vv

Switch the rows and columns - imagine it kinda swinging up/down a 90 degree angle

a square matrix whose entries not on the main diagonal are all zero

1 0

0 -1

Variables that are not governed by any quantifier (basically any column that doesn't have the leading entries in it see photos) (when writing the free variable in vector form, be sure to set it equal to t like x^3 = t)

number of nonzero in its ROW ECHELON FORM (if sys. is consistent, then the # of free variables equals -5x^2

1. Write augmented matrix of system of linear equations

2. Use elementary row operations to reduce augmented matrix to row echelon form.

3. rewrite equations using the REF form THEN Using back substitution, solve equivalent system that corresponds to row reduced matrix.

ANY MATRIX WITH MORE COLUMNS THAN ROWS IS LINEARLY DEPENDENT!!!

• Matrix A has eigenvalues 4 & -2 so

• Matrix A^-1 has eigenvalues ¼ and 1/-2 :))