# MA 237 Standards

Linear Algebra - 2018 Spring

Suggested homework exercises for each standard are taken from Linear Algebra with Applications (2nd ed) by Jeffrey Holt. The closest examples to exercises asked on assessments are given in bold.

An overview of course modules is available on this introductory handout.

Sample solutions for exercises representing each standard are available in this handout.

At the end of the course, each student should be able to…

• E1 [SysMat]: translate back and forth between a system of linear equations and the corresponding augmented matrix.
• §1.2 – 1-4, 11-18, 19-32
• E2 [Rref]: put a matrix in reduced row echelon form.
• §1.2 – 1-4, 11-18, 19-32
• E3 [SlvSys]: compute the solution set for a system of linear equations.
• §1.2 – 1-4, 11-18, 19-32
• V1 [VecPrp]: show why an example satisfies a given vector space property, but does not satisfy another given property.
• (study old quiz exercises and class activities)
• V2 [VecId]: list the eight defining properties of a vector space, infer which of these properties a given example satisfies, and thus determine if the example is a vector space.
• (study old quiz exercises and class activities)
• V3 [LinCmb]: determine if a Euclidean vector can be written as a linear combination of a given set of Euclidean vectors.
• §2.1 – 31-36
• §2.2 – 7-12
• V4 [Span]: determine if a set of Euclidean vectors spans $$\mathbb{R}^n$$.
• §2.2 – 21-28
• V5 [Subsp]: determine if a subset of $$\mathbb{R}^n$$ is a subspace or not.
• §4.1 – 1-16, 17-20
• S1 [LinInd]: determine if a set of Euclidean vectors is linearly dependent or independent.
• §2.3 – 1-12, 19-28
• S2 [BasVer]: determine if a set of Euclidean vectors is a basis of $$\mathbb{R}^n$$.
• V3 and S1 exercises
• §4.1 – 1-4, 23-28
• S3 [BasCmp]: compute a basis for the subspace spanned by a given set of Euclidean vectors.
• §4.2 – 5-16, 17-22
• S4 [Dim]: compute the dimension of a subspace of $$\mathbb{R}^n$$.
• §4.2 – 5-16, 17-22
• S5 [AbsVec]: solve exercises related to standards V3-S4 when posed in terms of polynomials or matrices.
• §7.1 – 13-14
• §7.2 – 1-24
• §7.3 – 7-10, 21-26
• S6 [BasSol]: find a basis for the solution set of a homogeneous system of equations.
• §1.2 – 20,25, 19-32 (replace constants with 0s)
• A1 [LinVer]: determine if a map between vector spaces of polynomials is linear or not.
• §3.1 – 13-20
• §9.1 – 5, Example 2 in section, Practice Problem 4a
• A2 [LinMat]: translate back and forth between a linear transformation of Euclidean spaces and its standard matrix, and perform related computations.
• §3.1 – 9-12,13,16-18
• A3 [InjSrj]: determine if a given Euclidean linear map is injective and/or surjective.
• §3.1 – 21-28
• A4 [KerImg]: compute a basis for the kernel and a basis for the image of a Euclidean linear map.
• §4.1 – 21-32 (note: the book calls the kernel of a transformation a “null space” for the standard matrix)
• M1 [MatMlt]: multiply matrices..
• §3.2 – A-E (find all possible products), 7-10
• M2 [InvVer]: determine if a square matrix is invertible or not.
• §3.3 – 1-6, 7-16
• M3 [InvCmp]: compute the inverse matrix of an invertible matrix.
• §3.3 – 1-6, 7-16
• G1 [RowOp]: represent a row operation as matrix multiplication, and compute how the operation affects the determinant.
• §5.2 – 7-14, 15-18
• G2 [Det]: compute the determinant of a square matrix.
• §5.1 – 11-18, 19-26
• G3 [EigVal]: find the eigenvalues of a $$2\times 2$$ matrix.
• §6.1 – 7-10, 21-24, 25-30
• G4 [EigVec]: find a basis for the eigenspace of a square matrix associated with a given eigenvalue.
• §6.1 – 11-14, 15-20, 21-30