Kernel and Transformation Concepts

The video tutorial explains a transformation from M22 to R3, defined by a specific matrix operation. It covers how to find a basis for the image or range of the transformation by expressing output vectors as linear combinations. The tutorial also details finding a basis for the kernel, focusing on input matrices that result in a zero vector output. The process involves breaking down matrices into linear combinations of simpler matrices.

What is the transformation T from M22 to R3 defined as?

• T of the 2x2 matrix abcd equals the vector a, b, c

• T of the 2x2 matrix abcd equals the vector a-b, a, a+b

• T of the 2x2 matrix abcd equals the vector a+b, a, a-b

• T of the 2x2 matrix abcd equals the vector b, a, d

How can the output vector a+b, a, a-b be expressed?

• As a sum of two vectors a, b, 0 and 0, 0, d

• As a sum of two vectors a, 0, 0 and 0, b, -b

• As a sum of two vectors a, a, a and b, 0, -b

• As a sum of two vectors a, b, c and d, 0, 0

What is the span of the set containing the vectors 1, 1, 1 and 1, 0, -1?

• The image or range of the transformation

• The kernel of the transformation

• The null space of the transformation

• The domain of the transformation

What must be true for the output vector to be the zero vector?

• a and b must both be zero

• a and c must both be zero

• b and d must both be zero

• c and d must both be zero

What is the form of the two by two matrices in the kernel?

• a, b, 0, 0

• 0, 0, c, d

• 0, 0, a, b

• c, d, 0, 0

How can the kernel be expressed?

• As a span of the matrices 1, 0, 0, 0 and 0, 1, 0, 0

• As a span of the matrices 0, 1, 0, 0 and 0, 0, 1, 0

• As a span of the matrices 0, 0, 1, 0 and 0, 0, 0, 1

• As a span of the matrices 1, 1, 1, 1 and 0, 0, 0, 0

What is a basis for the kernel of the transformation?

• The set containing the matrices 1, 0, 0, 0 and 0, 1, 0, 0

• The set containing the matrices 0, 1, 0, 0 and 0, 0, 1, 0

• The set containing the matrices 0, 0, 1, 0 and 0, 0, 0, 1

• The set containing the matrices 1, 1, 1, 1 and 0, 0, 0, 0

What does the transformation's image or range represent?

• The span of the vectors 1, 0, 0 and 0, 1, 0

• The span of the vectors 0, 0, 1 and 0, 0, 0

• The span of the vectors 0, 1, 0 and 0, 0, 1

• The span of the vectors 1, 1, 1 and 1, 0, -1

What is the significance of the kernel in this transformation?

• It represents the set of input matrices that map to the null vector

• It represents the set of input matrices that map to the zero vector

• It represents the set of input matrices that map to the identity matrix

• It represents the set of input matrices that map to the unit vector

What is the conclusion about the transformation's image and kernel?

• The image is the span of 0, 1, 0 and 0, 0, 1; the kernel is the span of 1, 1, 1 and 1, 0, -1

• The image is the span of 1, 0, 0 and 0, 1, 0; the kernel is the span of 0, 1, 0 and 0, 0, 1

• The image is the span of 0, 0, 1 and 0, 0, 0; the kernel is the span of 1, 0, 0 and 0, 1, 0

• The image is the span of 1, 1, 1 and 1, 0, -1; the kernel is the span of 0, 0, 1, 0 and 0, 0, 0, 1