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1. If A is a 3 x 4 matrix, then the kernel of A is a subspace of R^3.

I say this is false because since the kernel is a subspace of R^4, since there are 4 columns.

2. If A is a symmetric matrix, all of its eigenvalues are real and have multiplicity one.

Not exactly sure about this one.

3. If A is a Hermitian matrix, all of its eigenvectors are real.

Not sure about this one either.

4. If A is a square matrix with determinant zero, then its columns are linearly dependent.

I say this is true. Not sure why though. I'm thinking that there has to be a relationship between the columns since they will need to cancel out.

5. If A is a 4 x 3 matrix of rank 3, then its columns span R^3.

I say this is true because rank equals the number of pivots. Since there are 3 columns and 3 pivots, the basis is made of the 3 columns. Thus it spans R^3.