Linear Independence — Practice Problems and Worked Solutions

18 worked problems

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

1. 1.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly independent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=2$$
   2. $$\text{Linearly independent}$$
2. 2.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly dependent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=1$$
   2. $$\text{Linearly dependent}$$
3. 3.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly independent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=2$$
   2. $$\text{Linearly independent}$$
4. 4.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly dependent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=1$$
   2. $$\text{Linearly dependent}$$
5. 5.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly independent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=3$$
   2. $$\text{Linearly independent}$$
6. 6.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly dependent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=2$$
   2. $$\text{Linearly dependent}$$
7. 7.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly independent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=2$$
   2. $$\text{Linearly independent}$$
8. 8.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly dependent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=1$$
   2. $$\text{Linearly dependent}$$
9. 9.

   Classify the columns of the matrix.

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

   Answer: $\text{Linearly independent}$

   Step-by-step solution

   1. $$\mathrm{rank}(A)=2$$
   2. $$\text{Linearly independent}$$
10. 10.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly dependent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=1$$
    2. $$\text{Linearly dependent}$$
11. 11.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly independent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=3$$
    2. $$\text{Linearly independent}$$
12. 12.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly dependent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=1$$
    2. $$\text{Linearly dependent}$$
13. 13.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly independent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=2$$
    2. $$\text{Linearly independent}$$
14. 14.

    Classify the columns of the matrix.

    $$A=\left[\begin{array}{cc}4& 8\\ 1& 2\end{array}\right]$$

    Answer: $\text{Linearly dependent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=1$$
    2. $$\text{Linearly dependent}$$
15. 15.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly independent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=3$$
    2. $$\text{Linearly independent}$$
16. 16.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly dependent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=1$$
    2. $$\text{Linearly dependent}$$
17. 17.

    Classify the columns of the matrix.

    $$A=\left[\begin{array}{cc}5& 1\\ 2& 3\end{array}\right]$$

    Answer: $\text{Linearly independent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=2$$
    2. $$\text{Linearly independent}$$
18. 18.

    Classify the columns of the matrix.

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

    Answer: $\text{Linearly dependent}$

    Step-by-step solution

    1. $$\mathrm{rank}(A)=1$$
    2. $$\text{Linearly dependent}$$

More from Linear Algebra

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