Rank & Basis — Practice Problems and Worked Solutions

18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
2. 2.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$$

   Answer: $1$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}\right]$$
   3. $$\mathrm{rank}(A)=1$$
3. 3.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
4. 4.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{ccc}1& 2& 3\\ 2& 4& 6\\ 0& 1& 1\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 1\\ 0& 1& 1\\ 0& 0& 0\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
5. 5.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 0\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 2\\ 0& 1& 3\\ 0& 0& 0\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
6. 6.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{ccc}1& 2& 0\\ 0& 1& 1\\ 2& 5& 1\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& -2\\ 0& 1& 1\\ 0& 0& 0\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
7. 7.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{cc}2& 0\\ 0& 5\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
8. 8.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{cc}1& 3\\ 2& 6\end{array}\right]$$

   Answer: $1$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 3\\ 0& 0\end{array}\right]$$
   3. $$\mathrm{rank}(A)=1$$
9. 9.

   Find the dimension of the column space.

   $$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$$

   Answer: $2$

   Step-by-step solution

   1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
   2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]$$
   3. $$\mathrm{rank}(A)=2$$
10. 10.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{ccc}1& 1& 1\\ 2& 2& 2\\ 3& 3& 3\end{array}\right]$$

    Answer: $1$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 1& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=1$$
11. 11.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{ccc}1& 0& 3\\ 0& 1& 2\\ 0& 0& 1\end{array}\right]$$

    Answer: $3$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$
    3. $$\mathrm{rank}(A)=3$$
12. 12.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{cc}2& 4\\ 1& 2\end{array}\right]$$

    Answer: $1$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 2\\ 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=1$$
13. 13.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& 2\\ 1& 3& 3\end{array}\right]$$

    Answer: $2$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& -3\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=2$$
14. 14.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{ccc}3& 0& 0\\ 0& 0& 0\\ 0& 0& 7\end{array}\right]$$

    Answer: $2$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=2$$
15. 15.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{cc}1& 5\\ 0& 0\end{array}\right]$$

    Answer: $1$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 5\\ 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=1$$
16. 16.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{cccc}1& 0& 2& 0\\ 0& 1& 3& 0\end{array}\right]$$

    Answer: $2$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{cccc}1& 0& 2& 0\\ 0& 1& 3& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=2$$
17. 17.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{cc}2& 2\\ 3& 3\end{array}\right]$$

    Answer: $1$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=1$$
18. 18.

    Find the dimension of the column space.

    $$A=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 1& 1\end{array}\right]$$

    Answer: $2$

    Step-by-step solution

    1. $$\dim (\mathrm{Col}A)=\mathrm{rank}(A)$$
    2. $$\mathrm{rref}(A)=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 0& 0\end{array}\right]$$
    3. $$\mathrm{rank}(A)=2$$

More from Linear Algebra

• Vector Operations
• Dot Products
• Matrix Products
• Linear Systems
• Row Reduction
• Determinants
• Matrix Inverses
• Linear Independence
• Orthogonal Projections
• Linear Transformations
• Eigenvalues