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The mapping ? is called scalar multiplication and the binary operation ? is termed vector addition. 115. Page 2. 116 • Theory and Problems of Linear Algebra. properties of vectors play a fundamental role in linear algebra. In fact, in the next section these properties will be abstracted to define vector spaces. first course in Linear Algebra. The material is standard in that the subjects covered are Gaussian reduction, vector spaces, linear maps, determinants, A vector space is a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers) Theorem 3.4 – Checking linear independence in Rm. The following statements are equivalent for each m ? n matrix B. 1 The columns of B are linearly independent.Linear algebra is the mathematics of vector spaces and their subspaces. We will see that many questions about vector spaces can be reformulated as. the most important theorems in linear algebra. Theorem 2.3. All bases for a finite-dimenstional vector space have the same number of vectors. SchoolofMathematicsand. Statistics. 2. Matrices, row vectors and column vectors. Linear algebra is one of the most basic of all branches of mathematics. The.