Step-by-Step Guidance for Linear Algebra Midterm Exam

Q1. True/False and Short Answer Questions

Background

Topic: Linear Algebra Fundamentals

This section tests your understanding of basic concepts such as pivot positions, linear independence, and properties of matrices and vector spaces.

Key Terms:

• Pivot Position: A location in a matrix that corresponds to a leading entry in its row-echelon form.

• Linear Independence: A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others.

• Vector Equation: An equation involving vectors and scalars.

Step-by-Step Guidance

1. Review the definitions of pivot positions and linear independence.

2. Recall how to determine if a matrix has a pivot in every row or column by examining its row-echelon form.

3. For each statement, consider whether the property described is always true, sometimes true, or false based on the definitions.

4. For the numerical answers, think about how many free variables or solutions a system might have based on the matrix structure.

Try solving on your own before revealing the answer!

Final Answers:

• (a) True

• (b) True

• (c) True

• (d) False

• (e) False

• (f) False

• (g) 3

• (h) 8

These answers reflect the properties of matrices and systems of equations as discussed in class.

Q2. Solving Systems and Parametric Vector Form

Background

Topic: Solutions to Linear Systems

This question tests your ability to interpret the solution set of a linear system, write it in parametric vector form, and understand the implications of pivot positions.

Key Terms and Formulas:

• Parametric Vector Form: Expressing the solution to a system as a sum of vectors multiplied by parameters.

• Pivot Position: Determines if a system has a unique solution, infinitely many, or none.

• General Solution: $x = x_p + x_h$ where $x_p$ is a particular solution and $x_h$ is the homogeneous solution.

Step-by-Step Guidance

1. Examine the matrix to determine if there is a pivot in every row.

2. If not, recognize that the system may have infinitely many solutions or be inconsistent.

3. Write the general solution in parametric vector form using free variables as parameters.

4. For the particular solution, substitute values for the free variables to get a specific vector.

5. Express the solution as a sum of vectors multiplied by the parameters (free variables).

Try solving on your own before revealing the answer!

Final Answers:

• (a) No, because there is not a pivot position in every row.

• (b) $x = x_3 \begin{bmatrix} 2 \\ -1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_5 \begin{bmatrix} -4 \\ 5 \\ 0 \\ -3 \\ 1 \end{bmatrix}$

• (c) $x = \begin{bmatrix} -2 \\ 6 \\ 0 \\ 2 \\ 1 \end{bmatrix} + x_3 \begin{bmatrix} 2 \\ -1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + x_5 \begin{bmatrix} -4 \\ 5 \\ 0 \\ -3 \\ 1 \end{bmatrix}$

The solution is written in parametric vector form, showing the structure of the solution set.

Q3. Row Reduction and Solution Existence

Background

Topic: Row Reduction and Vector Equations

This question tests your ability to perform row reduction and determine if a vector equation has a solution based on pivot positions.

Key Terms:

• Row Reduction: The process of transforming a matrix to row-echelon form.

• Pivot Position: Indicates if a system is consistent.

• Vector Equation: $c_1 \vec{a}_1 + c_2 \vec{a}_2 + c_3 \vec{a}_3 = \vec{b}$

Step-by-Step Guidance

1. Perform row reduction on the given matrix to identify pivot positions.

2. Check if there is a pivot in every row; this indicates the system is consistent.

3. Interpret the result in terms of the vector equation: if consistent, a solution exists.

4. Set up the equation $c_1 \vec{a}_1 + c_2 \vec{a}_2 + c_3 \vec{a}_3 = \vec{b}$ and consider the implications of the pivots.

Try solving on your own before revealing the answer!

Final Answer:

Yes, because there is a pivot position in every row, so the corresponding vector equation $c_1 \vec{a}_1 + c_2 \vec{a}_2 + c_3 \vec{a}_3 = \vec{b}$ has a solution.

Q4. Row Reduction with Parameters

Background

Topic: Row Reduction with Parameters

This question tests your ability to perform row reduction on a matrix with parameters and determine conditions for consistency.

Key Terms:

• Row Reduction: Transforming a matrix to row-echelon form.

• Parameters: Variables such as $a$ and $b$ that affect the matrix structure.

• Consistency: Whether the system has a solution.

Step-by-Step Guidance

1. Write the matrix and identify the parameters $a$ and $b$.

2. Perform row reduction, noting how $a$ and $b$ affect the process.

3. Identify conditions on $a$ and $b$ that ensure a pivot in each row.

4. Set up equations based on the row-reduced form to solve for $a$ and $b$.

Try solving on your own before revealing the answer!

Final Answers:

• (a) Matrix as given.

• (b) $a \neq 3$

• (c) $b = 6$

These conditions ensure the matrix is consistent and has the required pivot positions.

Q5. Parametric Solution and Vector Equation

Background

Topic: Parametric Solutions and Vector Equations

This question tests your ability to express the solution to a system in parametric form and relate it to a vector equation.

Key Terms:

• Parametric Solution: Expressing the solution in terms of free variables.

• Vector Equation: Relating the solution to a linear combination of vectors.

Step-by-Step Guidance

1. Perform row reduction to identify free variables.

2. Express the solution vector in terms of the free variable(s).

3. Write the solution as a linear combination of vectors multiplied by the free variable(s).

4. Set up the vector equation based on the parametric form.

Try solving on your own before revealing the answer!

Final Answers:

• $x = x_3 \begin{bmatrix} -4 \\ 2 \\ 1 \end{bmatrix}$

• $-4v_1 + 2v_2 + v_3 = 0$

• $v_3 = 4v_1 - 2v_2$

The solution is expressed in parametric form and as a vector equation.

Q6. Matrix Properties: One-to-One and Onto

Background

Topic: Properties of Linear Transformations

This question tests your understanding of when a transformation is one-to-one or onto based on pivot positions in the matrix.

Key Terms:

• One-to-One: Each input maps to a unique output; requires a pivot in every column.

• Onto: Every possible output is covered; requires a pivot in every row.

• Matrix Dimensions: $n$ (number of columns), $m$ (number of rows).

Step-by-Step Guidance

1. Identify the dimensions of the matrix ($n$ and $m$).

2. Perform row reduction to check for pivot positions in columns and rows.

3. Determine if the transformation is one-to-one (pivot in every column) or onto (pivot in every row).

4. Set up the reasoning for each property based on the row-reduced form.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $n = 3$, $m = 2$

• (b) Not one-to-one, since there isn’t a pivot position in every column.

• (c) Onto, since there is a pivot position in every row.

These properties are determined by the structure of the matrix after row reduction.

Q7. Matrix Multiplication and Properties

Background

Topic: Matrix Multiplication

This question tests your ability to multiply matrices, determine when multiplication is defined, and compute products.

Key Terms:

• Matrix Multiplication: The process of multiplying two matrices.

• Defined Product: Multiplication is defined if the number of columns in the first matrix equals the number of rows in the second.

• Transpose: $A^T$ is the transpose of matrix $A$.

Step-by-Step Guidance

1. Check the dimensions of the matrices to determine if multiplication is defined.

2. Multiply matrices where possible, following the rules of matrix multiplication.

3. For $A^T B$, compute the product using the transpose of $A$ and matrix $B$.

4. Set up the multiplication and write out the resulting matrix.

Try solving on your own before revealing the answer!

Final Answers:

• (a) $Au = \begin{bmatrix} 7 \\ 10 \end{bmatrix}$

• (b) $AB$ is not defined.

• (c) $A^T B = \begin{bmatrix} 6 \\ -3 \\ 0 \\ 8 \\ 18 \\ -3 \end{bmatrix}$

Matrix multiplication is only defined when the dimensions match appropriately.

Q8. Linear Dependence and Transformations

Background

Topic: Linear Dependence and Linear Transformations

This question tests your understanding of how linear dependence is preserved under linear transformations.

Key Terms:

• Linear Dependence: A set of vectors is linearly dependent if there exist scalars, not all zero, such that their linear combination is zero.

• Linear Transformation: A function $T$ that preserves vector addition and scalar multiplication.

Step-by-Step Guidance

1. Recall the definition of linear dependence: $c_1 v_1 + c_2 v_2 = 0$ with at least one $c_i \neq 0$.

2. Apply the linear transformation $T$ to both sides: $T(c_1 v_1 + c_2 v_2) = T(0)$.

3. Use linearity: $T(c_1 v_1 + c_2 v_2) = c_1 T(v_1) + c_2 T(v_2)$.

4. Conclude that $\{T(v_1), T(v_2)\}$ is also linearly dependent.

Try solving on your own before revealing the answer!

Final Answer:

If $\{v_1, v_2\}$ is linearly dependent, then $\{T(v_1), T(v_2)\}$ is also linearly dependent, because the linear transformation preserves the dependence relation.