##### Exercise6

Determine whether the following statements are true or false and provide a justification for your response.

1. If the columns of a matrix $$A$$ form a basis for $$\real^m\text{,}$$ then $$A$$ is invertible.

2. There must be 125 vectors in a basis for $$\real^{125}\text{.}$$

3. If $$\bcal=\{\vvec_1,\vvec_2,\ldots,\vvec_n\}$$ is a basis of $$\real^m\text{,}$$ then every vector in $$\real^m$$ can be expressed as a linear combination of basis vectors.

4. The coordinates $$\coords{\xvec}{\bcal}$$ are the weights that form $$\xvec$$ as a linear combination of basis vectors.

5. If the basis vectors form the columns of the matrix $$C_{\bcal}\text{,}$$ then $$\coords{\xvec}{\bcal} = C_{\bcal}\xvec\text{.}$$

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