Understanding Transformations and Kernels

The video tutorial explains a transformation from R3 to P3, focusing on finding the basis for the image and kernel of the transformation. The image is defined as the span of x and x cubed, while the kernel consists of vectors in the form of zero zero c, indicating the span of the vector zero zero one. The tutorial emphasizes understanding the transformation's output and input to determine these bases.

What is the transformation T of vector abc defined as?

• ax squared plus bx

• ax plus bx squared

• ax plus bx cubed

• ax cubed plus bx

What is the image or range of the transformation?

• The set of all possible vectors in R3

• The set of all possible polynomials in P3

• The set of all possible matrices

• The set of all possible scalars

Which set forms a basis for the image of the transformation?

• The set containing ax and bx cubed

• The set containing x and x squared

• The set containing x and x cubed

• The set containing ax squared and bx

Why is the set containing ax and bx cubed not a suitable basis for the image?

• Because it ties the basis to specific values of a and b

• Because it includes non-linear terms

• Because it is not a complete set

• Because it is not in P3

What is the kernel of the transformation?

• The set of all input vectors that result in a non-zero polynomial

• The set of all output vectors that result in a non-zero vector

• The set of all input vectors that result in the zero polynomial

• The set of all output vectors that result in a zero vector

What must be true for the output to be the zero polynomial in the kernel?

• a and b must be zero

• c must be zero

• b must be zero

• a must be zero

What form do the vectors in the kernel take?

• Vectors in the form of (a, 0, c)

• Vectors in the form of (0, b, c)

• Vectors in the form of (0, 0, c)

• Vectors in the form of (a, b, 0)

What is the kernel of the transformation equal to?

• The span of the set containing the vector (0, 1, 0)

• The span of the set containing the vector (1, 1, 1)

• The span of the set containing the vector (0, 0, 1)

• The span of the set containing the vector (1, 0, 0)

What is a basis for the kernel of the transformation?

• Any scalar multiple of the vector (0, 1, 0)

• Any scalar multiple of the vector (1, 0, 0)

• Any scalar multiple of the vector (1, 1, 0)

• Any scalar multiple of the vector (0, 0, 1)

If c equals 1, what is a basis for the kernel?

• The vector (0, 0, 1)

• The vector (1, 0, 0)

• The vector (1, 1, 1)

• The vector (0, 1, 0)