**Candidates should be able to:**

- recall and use the axioms of a linear (vector) space (restricted to spaces of finite dimension over the field of real numbers only);
- understand the idea of linear independence, and determine whether a given set of Matrices and linear spaces is dependent or independent;
- understand the idea of the subspace spanned by a given set of Matrices and linear spaces;
- recall that a basis for a space is a linearly independent set of Matrices and linear spaces that spans the space, and determine a basis in simple cases;
- recall that the dimension of a space is the number of Matrices and linear spaces in a basis;
- understand the use of matrices to represent linear transformations from \(\mathbb{R}^n \longrightarrow \mathbb{R}^m\) ;
- understand the terms ‘column space’, ‘row space’, ‘range space’ and ‘null space’, and determine the dimensions of, and bases for, these spaces in simple cases;
- determine the rank of a square matrix, and use (without proof) the relation between the rank, the dimension of the null space and the order of the matrix;
- use methods associated with matrices and linear spaces in the context of the solution of a set of linear equations;
- evaluate the determinant of a square matrix and find the inverse of a non-singular matrix (2 × 2 and 3 × 3 matrices only), and recall that the columns (or rows) of a square matrix are independent if and only if the determinant is non-zero;
- understand the terms ‘eigenvalue’ and ‘eigenvector’, as applied to square matrices;
- find eigenvalues and eigenMatrices and linear spaces of 2 × 2 and 3 × 3 matrices (restricted to cases where the eigenvalues are real and distinct);
- express a matrix in the form \(\mathbf{QDQ}^{−1}\) , where \(\mathbf{D}\) is a diagonal matrix of eigenvalues and \(\mathbf{Q}\) is a matrix whose columns are eigenMatrices and linear spaces, and use this expression, e.g. in calculating powers of matrices.

**Matrices and linear spaces – Further Maths Paper 1 Videos Part 1 – 19**

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