University Linear Algebra — Inner Products and Orthogonality Worksheet

All 9 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Inner Product, Norm and Distance in n-Space

   In ${ℝ}^5$ with the dot product, let $\overrightarrow{u}=(1,-2,0,2,4)$ and $\overrightarrow{v}=(2,1,-2,0,4)$.

   1. Find $\overrightarrow{u}·\overrightarrow{v}$, $\Vert \overrightarrow{u}\Vert$ and $\Vert \overrightarrow{v}\Vert$.
   2. Find the distance $d(\overrightarrow{u},\overrightarrow{v})=\Vert \overrightarrow{u}-\overrightarrow{v}\Vert$.
   3. Find the unit vector in the direction of $\overrightarrow{v}$.
   4. Find $\cos \theta$, where $\theta$ is the angle between $\overrightarrow{u}$ and $\overrightarrow{v}$, and say whether $\theta$ is acute, right or obtuse.
2. Q2Inner Product, Norm and Distance in n-Space

   On ${ℝ}^4$ define $$⟨\overrightarrow{x},\overrightarrow{y}⟩=2x_1{y}_1+x_2{y}_2+3x_3{y}_3+x_4{y}_4,$$ where $\overrightarrow{x}=(x_1,x_2,x_3,x_4)$ and $\overrightarrow{y}=(y_1,y_2,y_3,y_4)$. Norm and distance are $\Vert \overrightarrow{x}\Vert =\sqrt{⟨\overrightarrow{x},\overrightarrow{x}⟩}$ and $d(\overrightarrow{x},\overrightarrow{y})=\Vert \overrightarrow{x}-\overrightarrow{y}\Vert$, computed with this inner product. Let $\overrightarrow{x}=(1,-1,1,2)$ and $\overrightarrow{y}=(0,3,1,-1)$.

   1. Explain why $⟨\overrightarrow{x},\overrightarrow{x}⟩>0$ for every $\overrightarrow{x}\ne \overrightarrow{0}$.
   2. Find $⟨\overrightarrow{x},\overrightarrow{y}⟩$, $\Vert \overrightarrow{x}\Vert$, $\Vert \overrightarrow{y}\Vert$ and $d(\overrightarrow{x},\overrightarrow{y})$. Compare $⟨\overrightarrow{x},\overrightarrow{y}⟩$ and $d(\overrightarrow{x},\overrightarrow{y})$ with the values the dot product gives.
   3. Show that $\overrightarrow{x}$ and $\overrightarrow{z}=(1,2,0,0)$ are orthogonal for this inner product but not for the dot product.
3. Q3Inner Products on Polynomial and Function Spaces

   On $P_2$, the polynomials of degree at most $2$, use the inner product $$⟨p,q⟩={\int }_{-1}^{1}p(x)q(x)dx,$$ with $\Vert p\Vert =\sqrt{⟨p,p⟩}$ and $d(p,q)=\Vert p-q\Vert$. Let $p(x)=1+2x$ and $q(x)=x-x^2$.

   1. Find $⟨p,q⟩$.
   2. Find $\Vert p\Vert$ and $\Vert q\Vert$.
   3. Find the distance $d(p,q)$.
   4. Find the value of the constant $a$ for which $r(x)=a+x^2$ is orthogonal to $q$.
4. Q4Orthogonal and Orthonormal Sets

   In ${ℝ}^4$ with the dot product, let ${\overrightarrow{v}}_1=(1,2,0,-1)$, ${\overrightarrow{v}}_2=(-2,1,-2,0)$ and ${\overrightarrow{v}}_3=(1,0,-1,1)$.

   1. Show that $\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3\}$ is an orthogonal set.
   2. Turn it into an orthonormal set.
   3. Is the set linearly independent? Is it a basis of ${ℝ}^4$? Justify both answers without row reducing.
5. Q5Orthogonal and Orthonormal Sets

   On $M_{2\times 2}$ use the inner product $⟨A,B⟩=a_{11}{b}_{11}+a_{12}{b}_{12}+a_{21}{b}_{21}+a_{22}{b}_{22}$, the sum of the products of corresponding entries. Let $$A_1=\left[\begin{array}{cc}1& 1\\ 3& -1\end{array}\right],A_2=\left[\begin{array}{cc}-3& -1\\ 1& -1\end{array}\right],A_3=\left[\begin{array}{cc}-1& 2\\ 0& 1\end{array}\right].$$

   1. Show that $\{A_1,A_2,A_3\}$ is an orthogonal set.
   2. Turn it into an orthonormal set.
   3. Is $\{A_1,A_2,I_2\}$ an orthogonal set?
6. Q6Coordinates Relative to an Orthogonal Basis

   In ${ℝ}^4$ with the dot product, let $ℬ=\{{\overrightarrow{b}}_1,{\overrightarrow{b}}_2,{\overrightarrow{b}}_3,{\overrightarrow{b}}_4\}$ with $${\overrightarrow{b}}_1=(3,1,1,-1),{\overrightarrow{b}}_2=(-1,-1,1,-3),{\overrightarrow{b}}_3=(-1,2,1,0),{\overrightarrow{b}}_4=(0,-1,2,1).$$

   1. Show that $ℬ$ is an orthogonal basis of ${ℝ}^4$.
   2. Find $[\overrightarrow{x}{]}_{ℬ}$ for $\overrightarrow{x}=(5,1,2,-3)$ without solving a linear system.
7. Q7Coordinates Relative to an Orthogonal Basis

   On $P_2$ use the inner product $⟨p,q⟩=p(0)q(0)+p(1)q(1)+p(2)q(2)$. Let $ℬ=\{1,x-1,3x^2-6x+1\}$.

   1. Show that $ℬ$ is an orthogonal basis of $P_2$ for this inner product.
   2. Find the coordinates of $p(x)=2x^2-x+3$ relative to $ℬ$ by computing inner products, and check your answer.
8. Q8The Orthogonal Complement of a Subspace

   In ${ℝ}^5$ with the dot product, let $W=span\{(1,0,2,-1,1),(0,1,-1,2,0)\}$. The orthogonal complement is $W^{⟂}=\{\overrightarrow{x}\in {ℝ}^5\mid \overrightarrow{x}·\overrightarrow{w}=0\text{ for every }\overrightarrow{w}\in W\}$. Find a basis for $W^{⟂}$, and check that $\dim W+\dim W^{⟂}=5$.

9. Q9Synthesis — drawing on several topics in this unit

   In ${ℝ}^4$ with the dot product, let ${\overrightarrow{v}}_1=(2,1,-1,1)$, ${\overrightarrow{v}}_2=(-1,-1,-2,1)$, ${\overrightarrow{v}}_3=(0,-1,2,3)$ and $W=span\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3\}$.

   1. Show that $\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3\}$ is an orthogonal set, and find $\dim W$.
   2. Find a basis for $W^{⟂}$, the set of vectors orthogonal to every vector of $W$.
   3. Let $\overrightarrow{u}$ be the unit vector in $W^{⟂}$ whose last component is positive. Find the coordinates of $\overrightarrow{x}=(3,0,1,2)$ relative to the orthogonal basis $\{{\overrightarrow{v}}_1,{\overrightarrow{v}}_2,{\overrightarrow{v}}_3,\overrightarrow{u}\}$ of ${ℝ}^4$.

The 6 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Linear Algebra bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Inner Products and Orthogonality notes cover  About the Inner Products and Orthogonality unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Norm, distance and angle in n-space

Q1 is the dot product in five coordinates, and everything else follows from it: the norm is the square root of a vector's dot product with itself, the distance between two vectors is the norm of their difference, and the angle comes from the dot product divided by the two norms.

An inner product is a choice

The dot product is one inner product among many. Q2 defines a weighted one on four-space, with different positive weights on the four coordinate products. Part (a) asks why it still gives every non-zero vector a positive length: write out the inner product of x with itself and look at what kind of terms it adds.

Part (c) makes the point sharper. Orthogonal means inner product zero, so whether two vectors are orthogonal depends on which inner product you use. Compute both and say which is which.

Inner products on polynomials

Q3 uses an integral from −1 to 1 of the product of two polynomials as the inner product on quadratics. Each part is an integral of a polynomial: multiply out, integrate term by term, and evaluate.

Part (c) is a distance, so it is the norm of the difference: subtract the two polynomials first, then take the inner product of the difference with itself. Part (d) sets an inner product to zero and solves for the unknown constant — one linear equation.

Orthogonal and orthonormal sets

A set is orthogonal when every pair of distinct vectors in it has inner product zero, and orthonormal when, in addition, every vector has norm 1. Q4 asks you to check a set of three in four-space, then normalise it.

Q4(c) asks two questions without row reducing. Orthogonal sets are independent as long as none of their vectors is zero — pair a vanishing combination with each vector in turn and every weight is forced to zero. Whether they form a basis is then a question of counting, as in the Basis set.

Q5 is the same work on 2 × 2 matrices, with the inner product that multiplies corresponding entries and adds. Part (c) replaces one matrix by the identity and asks again — compute the pairs that involve the new matrix.

Coordinates relative to an orthogonal basis

This is the practical reason orthogonal bases matter. When a basis is orthogonal, each coordinate is one ratio of inner products, with no system to solve.

Pair both sides of x = c₁b₁ + ⋯ + cₙbₙ with one bᵢ in the inner product: every other term vanishes by orthogonality, leaving one equation in one unknown. Q6 asks you first to show the basis is orthogonal — six pairs, and then a count — and then to find coordinates with that formula.

Q7 is the same idea on quadratics with an inner product built from values at three points. Part (a) is six inner products — three pairs to show orthogonality plus the three norms you will need in part (b) — and the count that makes three independent quadratics a basis. Part (b) applies the formula and then asks for the check: rebuild p from its coordinates.

The orthogonal complement

The orthogonal complement of a subspace W is every vector orthogonal to all of W. Q8 gives W as the span of two vectors in five-space. A vector is orthogonal to all of W exactly when it is orthogonal to each spanning vector, so the complement is the solution set of two equations — the null space of the matrix whose rows are the spanning vectors.

Solve, read off a basis with one vector per free variable, and check that the two dimensions add up to five. That check is the rank theorem from the rank set, in a new setting.

The synthesis question

Q9 puts the set together in four-space. Part (a) is Q4: check the three pairs, then use the independence of orthogonal sets without a zero vector to read off the dimension. Part (b) is Q8: the complement is the null space of the matrix with the three vectors as rows. Part (c) normalises the complement's basis vector — the sign condition picks one of the two unit vectors — and then uses Q6's formula with four orthogonal basis vectors, remembering that for the unit vector the denominator is 1.

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Getting the most out of it

Write down the inner product before every computation

Several inner products appear in this set. Before computing a norm, a distance or an angle, write down which one the question uses. A correct dot product is a wrong answer when the question asked for a weighted inner product.

Keep norms squared until the end

Most formulas use ‖v‖², not ‖v‖. Carry the squared norm and take a square root only when the question asks for a length or a unit vector.

Count pairs

For an orthogonal-set question, list every pair before you compute. For n vectors there are n(n − 1)/2 pairs, and a missed pair is a missing mark.

Rebuild the vector

After finding coordinates relative to an orthogonal basis, add the weighted basis vectors back up. The result must be the vector you started with.

Common questions