Chapter 3: Problem 14
(a) Prove that any two-dimensional vector can be written as a linear combination of \(\mathbf{i}+\mathbf{j}\) and \(\mathbf{i}-\mathbf{j}\). (b) Prove that if the vectors \(\mathbf{x}=x_{1} \mathbf{i}+x_{2} \mathbf{j}\) and \(\mathbf{y}=y_{1} \mathbf{i}+y_{2} \mathbf{j}\) are linearly independent, then any vector \(\mathbf{z}=z_{1} \mathbf{i}+z_{2} \mathbf{j}\) can be expressed as a linear combination of \(\mathbf{x}\) and \(\mathbf{y} .\) Note that if \(\mathbf{x}\) and \(\mathbf{y}\) are linearly independent, then \(x_{1} y_{2}-x_{2} y_{1} \neq 0 .\) Why?
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.