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Linear Algebra

Linear Algebra

Linear Algebra


1.     (a)  Complete the definition:  Vectors  , . . . ,  are linearly independent if...

Answer:  ...no vi  is in the span of v1 , . . . , vi1 .

OR ...they  do not satisfy any nontrivial linear relation.

(b)  Suppose A is an m × n matrix whose columns are linearly independent.

What is the nullity of A?

Answer:  If the columns are linearly  independent,  the rank  of A is equal to n

(the dimension of the span of the columns), so the nullity equals zero.


2. Let A denote the matrix


Let T : R2 R2 be the linear transformation given by T ( x) = A x. (a)  (5 points) Describe T geometrically.

 For any angle θ, the matrix for counterclockwise rotation by θ is

Since cos   and sin




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