Answer Exercise 3.3 .1 for (a) ( 2 -1 ),( 1 -2 ), (b) ( 1 0

step 1 For this question, we are given that the two matrices on the board are equal to one another, and

Answer Exercise 3.3 .1 for (a) ( 2 -1 ),( 1 ...

Answer Exercise 3.3 .1 for (a) $\left(\begin{array}{r}2 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ -2\end{array}\right)$, (b) $\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)$, , (c) $\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ -3\end{array}\right)$.

Answer Exercise 3.3 .1 for (a) $\left(\begin{array}{r}2 \\ -1\end{array}\right),\left(\begin{array}{r}1 \\ -2\end{array}\right)$,
(b) $\left(\begin{array}{r}1 \\ 0 \\ -1\end{array}\right),\left(\begin{array}{r}-1 \\ 1 \\ 0\end{array}\right)$,
, (c) $\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right),\left(\begin{array}{r}2 \\ -1 \\ -3\end{array}\right)$.

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Key Concepts

Linear Independence

Linear independence is a core concept in linear algebra that determines whether a set of vectors can be expressed as nontrivial linear combinations of each other. In essence, a set of vectors is linearly independent if the only solution to the equation c?v? + c?v? + ... + c?v? = 0 is when all the coefficients c?, c?, ..., c? are zero. This concept is fundamental in understanding the structure and dimensions of vector spaces.

Systems of Linear Equations

Testing for linear independence typically involves setting up a system of linear equations by equating a linear combination of the vectors to the zero vector. Solving this homogeneous system reveals whether nontrivial solutions exist. If only the trivial solution (all coefficients zero) is found, the vectors are linearly independent; if a nontrivial solution exists, the vectors are linearly dependent.

Determinant Method

In cases involving two vectors in a two-dimensional space, the determinant of a 2×2 matrix formed by these vectors can be used as a criterion for linear independence. A nonzero determinant indicates that the vectors are linearly independent, as the area of the parallelogram they span is nonzero, whereas a zero determinant signals that the vectors are collinear and thus linearly dependent.

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