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.
  • B𝐴 is a projection matrix.
  • C𝐴 has 𝑚 linearly independent eigenvectors.
  • DThe kernal of 𝐴 is {}0.
  • E𝐴 is idempotent.

Q2:

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

  • A⎧⎪⎪⎨⎪⎪⎩⎛⎜⎜⎜⎜⎝−12−2100⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝−12−1010⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎝−11001⎞⎟⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭
  • B⎧⎪⎪⎨⎪⎪⎩⎛⎜⎜⎜⎜⎝100−2−12⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝010−1−12⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎝0011−1⎞⎟⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭
  • C⎧⎪⎨⎪⎩⎛⎜⎜⎜⎝100−1−1⎞⎟⎟⎟⎠,⎛⎜⎜⎜⎝010−1−1⎞⎟⎟⎟⎠,⎛⎜⎜⎜⎝0011−1⎞⎟⎟⎟⎠⎫⎪⎬⎪⎭
  • D⎧⎪⎪⎨⎪⎪⎩⎛⎜⎜⎜⎜⎝−2−12100⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎜⎝−1−12010⎞⎟⎟⎟⎟⎠,⎛⎜⎜⎜⎝1−1001⎞⎟⎟⎟⎠⎫⎪⎪⎬⎪⎪⎭
  • E⎧⎪⎨⎪⎩⎛⎜⎜⎜⎝−1−1100⎞⎟⎟⎟⎠,⎛⎜⎜⎜⎝−1−1010⎞⎟⎟⎟⎠,⎛⎜⎜⎜⎝1−1001⎞⎟⎟⎟⎠⎫⎪⎬⎪⎭

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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