Worksheet: Image and Kernel of Linear Transformation

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In this worksheet, we will practice finding the image and basis of the kernel of a linear transformation.

Q1:

Suppose that the columns of an 𝑚×𝑚 matrix 𝐴 are linearly independent. Then, which of the following statements is always true?

  • AThe determinant of 𝐴 is 1.
  • B 𝐴 is a projection matrix.
  • C 𝐴 has 𝑚 linearly independent eigenvectors.
  • DThe kernal of 𝐴 is {}0.
  • E 𝐴 is idempotent.

Q2:

Find a basis for ker(𝐴) , where 𝐴=⎛⎜⎜⎝12321021121443302112⎞⎟⎟⎠.

  • A ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ − 1 2 − 2 1 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ − 1 2 − 1 0 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ − 1 1 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
  • B ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 0 0 − 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 1 0 − 1 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 1 1 − 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
  • C ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ 1 0 0 − 1 − 1 ⎞ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ 0 1 0 − 1 − 1 ⎞ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ 0 0 1 1 − 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎬ ⎪ ⎭
  • D ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ − 2 − 1 2 1 0 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ − 1 − 1 2 0 1 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ 1 − 1 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
  • E ⎧ ⎪ ⎨ ⎪ ⎩ ⎛ ⎜ ⎜ ⎜ ⎝ − 1 − 1 1 0 0 ⎞ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ − 1 − 1 0 1 0 ⎞ ⎟ ⎟ ⎟ ⎠ , ⎛ ⎜ ⎜ ⎜ ⎝ 1 − 1 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎬ ⎪ ⎭

Q3:

Fill in the blank. A linear map 𝑇∈𝐿(𝑉,𝑊) is , if and only if 𝑇 is injective and surjective.

  • Ainvertible
  • Bone-to-one
  • Ccontinuous
  • Dsingular

Q4:

Let 𝑇∈𝐿ℝ be such that 𝑇𝑥𝑦=𝑦−𝑥𝑥𝑦∈ℝ.forall Then, 𝑇 is .

  • Aa surjective linear map
  • Ba zero map
  • Ca vector space
  • Dan identity map

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