Understanding Kernel of a Transformation

The video tutorial explains a transformation from R3 to R2 and how to find a non-zero vector in the kernel of the transformation. It begins by defining the transformation and the kernel, then sets up the problem of finding a non-zero vector in the kernel. The tutorial proceeds to solve the system of homogeneous equations using an augmented matrix and parameterizes the solution for vector X. Finally, it identifies a specific non-zero vector in the kernel by setting a parameter value.

What is the transformation T from R3 to R2 defined as?

• T(x1, x2, x3) = (x2 - x1, x3 + x1)

• T(x1, x2, x3) = (x1 + x2, x3 - x2)

• T(x1, x2, x3) = (x3 + x2, x1 - x2)

• T(x1, x2, x3) = (x1 - x3, x2 + x3)

What is the kernel of a transformation?

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

• The set of all output vectors that are equal to the input vectors

• The set of all input vectors that map to the zero vector

• The set of all input vectors that map to any vector

What is the zero vector in R2?

• (1, 1)

• (0, 1)

• (1, 0)

• (0, 0)

What is the first equation derived from the transformation?

• x1 - x3 = 0

• x3 + x2 = 0

• x3 - x2 = 0

• x1 + x2 = 0

What is the second equation derived from the transformation?

• x1 - x2 = 0

• x2 + x3 = 0

• x2 - x3 = 0

• x1 + x3 = 0

How can the solution for vector x be parameterized?

• x1 = -t, x2 = t, x3 = t

• x1 = t, x2 = t, x3 = t

• x1 = t, x2 = t, x3 = -t

• x1 = t, x2 = -t, x3 = t

What is the form of vectors in the kernel of T?

• (1, 0, -1)

• (0, 1, 1)

• (1, 1, -1)

• (1, -1, 1)

What is a non-zero vector in the kernel of T when t = 1?

• (0, 0, 0)

• (1, -1, -1)

• (1, 1, -1)

• (1, 1, 1)

Why can't t be zero when finding a non-zero vector in the kernel?

• Because it would result in a non-zero vector

• Because it would result in a zero vector

• Because it would result in a vector outside R3

• Because it would result in an undefined vector

What does the vector (1, 1, -1) represent in the context of the kernel?

• A vector that does not satisfy the transformation

• A vector outside the kernel

• A zero vector in the kernel

• A non-zero vector in the kernel