Mathematics II · Unit 1 · 5 hrs
Linear Equations in Linear Algebra
Exam-focused notes for Linear Equations in Linear Algebra (Mathematics II, MTH168): what the TU syllabus asks and how it has actually been tested, with 16 solved past questions from this unit.
What this unit covers
- System of linear equations
- Row reduction and Echelon forms
- Vector equations
- The matrix equations Ax = b
- Applications of linear system
- Linear independence
Row reduction and Echelon forms
Define system of linear equations. When a system of equations is consistent? Make echelon form to solve:
$-2a - 3b + 4c = 5$
$b - 2c = 4$
$a + 3b - c = 2$
[10]
System of 3 equations in 3 unknowns: $$-2a - 3b + 4c = 5 \quad \cdots (1)$$ $$b - 2c = 4 \quad \cdots (2)$$ $$a + 3b - c = 2 \quad \cdots (3)$$ --- A linear equation in $n$ variables has the form: $$a1x1 + a2x2 + \cdots + anxn = b$$ where $a1, a2, \ldots, a...
Full solved answer →Reduce the system of equations into echelon form and solve:
$$x_1 - 2x_2 - x_3 + 3x_4 = 0$$ $$-2x_1 + 4x_2 + 5x_3 - 5x_4 = 3$$ $$3x_1 - 6x_2 - 6x_3 + 8x_4 = 2$$
[10]
System of equations: $$x1 - 2x2 - x3 + 3x4 = 0$$ $$-2x1 + 4x2 + 5x3 - 5x4 = 3$$ $$3x1 - 6x2 - 6x3 + 8x4 = 2$$ Augmented matrix: $$[A \mid b] = \begin{bmatrix} 1 & -2 & -1 & 3 & \mid & 0 \\ -2 & 4 & 5 & -5 & \mid & 3 \\ 3 & -6 & -6 & 8 & \mid & 2 \end{bmatri...
Full solved answer →Change into reduced echelon form of the matrix $$\begin{pmatrix} 0 & 3 & -6 \ 3 & -1 & 8 \ 3 & -9 & 12 \end{pmatrix}$$ [5]
$$A = \begin{pmatrix} 0 & 3 & -6 \\ 3 & -1 & 8 \\ 3 & -9 & 12 \end{pmatrix}$$ All entries present. Task: reduce to RREF. --- $$\begin{pmatrix} 3 & -1 & 8 \\ 0 & 3 & -6 \\ 3 & -9 & 12 \end{pmatrix}$$ Row 3: $(3-3,\ -9-(-1),\ 12-8) = (0, -8, 4)$ $$\begin{pmat...
Full solved answer →Linear independence
Determine the column of the matrix A are linearly independent where $A = \begin{bmatrix} 3 & -3 & 6 \ 0 & 2 & 4 \ 0 & 3 & 0 \end{bmatrix}$ [5]
$$A = \begin{bmatrix} 3 & -3 & 6 \\ 0 & 2 & 4 \\ 0 & 3 & 0 \end{bmatrix}$$ Columns: $\mathbf{a1} = \begin{bmatrix}3\\0\\0\end{bmatrix}$, $\mathbf{a2} = \begin{bmatrix}-3\\2\\3\end{bmatrix}$, $\mathbf{a3} = \begin{bmatrix}6\\4\\0\end{bmatrix}$ The columns ar...
Full solved answer →Determine the column of the matrix A are linearly independent, where $A = \begin{bmatrix} -2 & 8 & -1 \ 0 & 0 & 0 \ 0 & -5 & 3 \end{bmatrix}$ [5]
$$A = \begin{bmatrix} -2 & 8 & -1 \\ 0 & 0 & 0 \\ 0 & -5 & 3 \end{bmatrix}$$ Columns: - $\mathbf{a}1 = \begin{bmatrix} -2 \\ 0 \\ 0 \end{bmatrix}$, $\mathbf{a}2 = \begin{bmatrix} 8 \\ 0 \\ -5 \end{bmatrix}$, $\mathbf{a}3 = \begin{bmatrix} -1 \\ 0 \\ 3 \end{...
Full solved answer →Determine the column of the matrix $A$ are linearly independent, where $A = \begin{bmatrix} 0 & 1 & 4 \ 1 & 2 & -1 \ 5 & 8 & 0 \end{bmatrix}$ [5]
$$A = \begin{bmatrix} 0 & 1 & 4 \\ 1 & 2 & -1 \\ 5 & 8 & 0 \end{bmatrix}$$ Task: determine whether the columns of $A$ are linearly independent. The columns are linearly independent iff $A\mathbf{x}=\mathbf{0}$ has only the trivial solution. A quick check is...
Full solved answer →Are vectors $v_1$, $v_2$, and $v_3$ linearly independent? Justify.
$$v_1 = \begin{bmatrix} 1 \ 4 \ 0 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 10 \ 2 \ 1 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -5 \ 0 \ 6 \end{bmatrix}$$
[5]
$$v1 = \begin{bmatrix} 1 \\ 4 \\ 0 \end{bmatrix},\quad v2 = \begin{bmatrix} 10 \\ 2 \\ 1 \end{bmatrix},\quad v3 = \begin{bmatrix} -5 \\ 0 \\ 6 \end{bmatrix}$$ The vectors are linearly independent iff the only solution to $c1 v1 + c2 v2 + c3 v3 = 0$ is the t...
Full solved answer →Vector equations
When two column vectors in $\mathbb{R}^2$ are equal? Give an example. Compute $u + 3v$, $u - 2v$, where $u = \begin{bmatrix} 1 \ -3 \ 2 \end{bmatrix}, \quad v = \begin{bmatrix} 1 \ -1 \ 3 \end{bmatrix}$ [5]
Given data: $$\mathbf{u} = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}$$ Tasks: 1. State the condition for equality of two column vectors in $\mathbb{R}^2$, with an example. 2. Compute $\mathbf{u...
Full solved answer →When two column vectors in $\mathbb{R}^2$ are equal? Give an example. Compute $u + 3v$, $u - 2v$, where $u = \begin{bmatrix} 1 \ -3 \ 2 \end{bmatrix}$, $v = \begin{bmatrix} 1 \ -1 \ 3 \end{bmatrix}$ [5]
$$\mathbf{u} = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}$$ Required: definition of vector equality with example; compute $\mathbf{u} + 3\mathbf{v}$ and $\mathbf{u} - 2\mathbf{v}$. Note: The hea...
Full solved answer →When two column vectors in $\mathbb{R}^2$ are equal? Give an example. Compute $u + 3v$, $u - 2v$, where $u = \begin{bmatrix} 1 \ -3 \ 2 \end{bmatrix}$, $v = \begin{bmatrix} 1 \ -1 \ 3 \end{bmatrix}$ [5]
$$\mathbf{u} = \begin{bmatrix} 1 \\ -3 \\ 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 1 \\ -1 \\ 3 \end{bmatrix}$$ Required: - Condition for equality of two column vectors in $\mathbb{R}^2$ with an example - Compute $\mathbf{u} + 3\mathbf{v}$ and $\...
Full solved answer →For what value of h will y be in span ${v_1, v_2, v_3}$ if $v_1, v_2, v_3$ and y are given as:
$$v_1 = \begin{bmatrix} 1 \ -1 \ -2 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 5 \ -4 \ -7 \end{bmatrix}, \quad v_3 = \begin{bmatrix} -3 \ 1 \ 0 \end{bmatrix}, \quad \text{and } y = \begin{bmatrix} -4 \ 3 \ h \end{bmatrix}$$
[5]
$$v1 = \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix}, \quad v2 = \begin{bmatrix} 5 \\ -4 \\ -7 \end{bmatrix}, \quad v3 = \begin{bmatrix} -3 \\ 1 \\ 0 \end{bmatrix}, \quad y = \begin{bmatrix} -4 \\ 3 \\ h \end{bmatrix}$$ $y \in \text{Span}\{v1, v2, v3\}$ if an...
Full solved answer →System of linear equations
When a linear system of equation is consistent? Find the values of h and k for which the system is consistent: $$2x_1 - x_2 = h$$ $$-6x_1 + 3x_2 = k$$ [5]
A system of linear equations is consistent if it has at least one solution (either a unique solution or infinitely many solutions). It is inconsistent if it has no solution. Using the augmented matrix approach, a system is consistent if and only if the redu...
Full solved answer →Define system of linear equations. When a system of equation is consistent? Determine if the system is consistent:
$$-2x_1 - 3x_2 + 4x_3 = 5$$ $$x_2 - 2x_3 = 4$$ $$x_1 + 3x_2 - x_3 = 2$$
[10]
A system of linear equations is a collection of one or more linear equations involving the same set of variables. A single linear equation has the form: $$a1x1 + a2x2 + \cdots + anxn = b$$ where $a1, \ldots, an, b$ are constants and $x1, \ldots, xn$ are unk...
Full solved answer →When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:
$x - 2y = 5$
$-x + y + 5z = 2$
$y + z = 0$
[10]
System of equations: $$x - 2y = 5 \quad \cdots (1)$$ $$-x + y + 5z = 2 \quad \cdots (2)$$ $$y + z = 0 \quad \cdots (3)$$ Unknowns: $x, y, z$ (3 unknowns, 3 equations). Note: equation (1) has no $z$ term (coefficient 0) and equation (3) has no $x$ term (coef...
Full solved answer →When a system of linear equation is consistent and inconsistent? Give an example for each. Test the consistency and solve:
$x + y + z = 4$
$x + 2y + 2z = 2$
$2x + 2y + z = 5$
[10]
System of equations: $$x + y + z = 4 \quad \cdots (1)$$ $$x + 2y + 2z = 2 \quad \cdots (2)$$ $$2x + 2y + z = 5 \quad \cdots (3)$$ --- Consistent System: A system of linear equations is consistent if it has at least one solution (either a unique solution or ...
Full solved answer →What is a system of linear equations? When the system is consistent? Find the condition on g, h, k that makes the system consistent.
$$x_1 - 4x_2 + 7x_3 = g$$ $$3x_2 - 5x_3 = h$$ $$-2x_1 + 5x_2 - 9x_3 = k$$
[10]
System of equations: $$x1 - 4x2 + 7x3 = g \quad (1)$$ $$3x2 - 5x3 = h \quad (2)$$ $$-2x1 + 5x2 - 9x3 = k \quad (3)$$ Coefficient matrix and augmented matrix: $$A = \begin{bmatrix} 1 & -4 & 7 \\ 0 & 3 & -5 \\ -2 & 5 & -9 \end{bmatrix}, \qquad [A \mid b] = \b...
Full solved answer →Make Unit 1 stick
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