Worksheet: Row Space and Column Space

In this worksheet, we will practice defining the row and column spaces of a matrix and finding the basis and dimensions of these vector spaces.

Q1:

What is the row space of matrix 𝐴 of size 𝑚×𝑛 a subspace of?

  • A𝑅
  • B𝑅 and 𝑅
  • C𝑅

Q2:

Consider the matrix 𝐴 of size 𝑚×𝑛.

Dimension of the row space of 𝐴=𝐴=dimensionofthecolumnspaceof.

  • Arank of 𝐴
  • B𝑚+𝑛
  • Cdeterminant of 𝐴
  • Dtrace of 𝐴
  • E𝑚×𝑛

Q3:

The column space of matrix 𝐴 of size 𝑚×𝑛 is a subspace of .

  • A𝑅
  • B𝑅
  • C𝑅 and 𝑅

Q4:

The column space of 𝐴=therowspaceof.

  • A𝐴
  • B𝐴
  • C𝐴
  • D𝐴×𝐴

Q5:

For matrix 𝐴 of size 𝑚×𝑛, which of the following statements is true?

  1. Rank = minimum of (𝑚,𝑛).
  2. Column space is a subspace of 𝑅.
  3. Dimension of column space = dimension of row space.
  4. Row space is a subspace of 𝑅.
  • AII only
  • BI, II, and III
  • CI, III, and IV
  • DII and III
  • EIII and IV

Q6:

What is the dimension of the row space of the matrix 121321360237833?

Q7:

Find the basis for the column space of the matrix 210312.

  • A11
  • B02
  • C23,02
  • D11,02
  • E23,11

Q8:

Find the basis for the row space of the following matrix:121312124623.

  • A1212
  • B1213,1212
  • C1213,4623
  • D1212,4623
  • E1213,1212,4623

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