Understanding Kernel of a Transformation

The video tutorial explains a transformation from P2 to R2, focusing on finding a non-trivial element of the kernel. The kernel is defined as the set of input polynomials that result in a zero vector output. The tutorial analyzes polynomial inputs, emphasizing the need for them to be in factored form to find the roots. It concludes by identifying non-trivial elements of the kernel, highlighting that any scalar multiple of a specific polynomial form, except zero, is valid.

What is the transformation T from P2 to R2 described as?

• T of p equals the vector p of 0, p of 5

• T of p equals the vector p of 1, p of 3

• T of p equals the vector p of 2, p of 4

• T of p equals the vector p of 3, p of 1

What is the kernel of a transformation?

• A set of all output vectors that are non-zero

• A set of all input polynomials that result in a zero vector

• A set of all input vectors that result in a non-zero vector

• A set of all output polynomials that are non-zero

What must be true for the output vector in the kernel?

• The output vector must be a scalar multiple of the input

• The output vector must be a unit vector

• The output vector must be non-zero

• The output vector must be the zero vector

What form can any degree two polynomial be written in?

• a t plus b

• a t squared plus b

• a t squared plus b t plus c

• a t cubed plus b t squared plus c

What is the factored form of a degree two polynomial?

• a times the quantity t minus r1 times the quantity t minus r2

• a times the quantity t plus r1 minus r2

• a times the quantity t plus r1 times the quantity t plus r2

• a times the quantity t minus r1 plus r2

What are the roots of the polynomial for the kernel?

• r1 equals 0 and r2 equals 5

• r1 equals 1 and r2 equals 3

• r1 equals 3 and r2 equals 1

• r1 equals 2 and r2 equals 4

What form must the polynomials in P2 be for the kernel?

• a times the quantity t minus 0 times the quantity t minus 5

• a times the quantity t minus 1 times the quantity t minus 3

• a times the quantity t minus 2 times the quantity t minus 4

• a times the quantity t minus 3 times the quantity t minus 1

What is a non-trivial element of the kernel?

• p of t equals the quantity t minus 3 times the quantity t minus 1

• p of t equals the quantity t minus 0 times the quantity t minus 5

• p of t equals the quantity t minus 1 times the quantity t minus 3

• p of t equals the quantity t minus 2 times the quantity t minus 4

What condition must 'a' satisfy to avoid the trivial solution?

• a must be zero

• a must be greater than zero

• a must be less than zero

• a must not be zero

What is the significance of scalar multiples in the kernel?

• They form the trivial solution

• They only apply to zero vectors

• They are irrelevant to the kernel

• They form the non-trivial solution