Examples of Standard Bases in R^2, R^3, R^n, and Polynomials

📚 Quick Study Guide

• 🔢 A basis for a vector space is a set of linearly independent vectors that span the entire space.
• 📍 The standard basis is a specific, convenient basis that is commonly used.
• 📈 In $R^2$, the standard basis is usually given by the vectors $\{(1, 0), (0, 1)\}$. Any vector $(x, y)$ in $R^2$ can be written as a linear combination of these basis vectors: $(x, y) = x(1, 0) + y(0, 1)$.
• ✨ In $R^3$, the standard basis is typically given by the vectors $\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\}$. A vector $(x, y, z)$ in $R^3$ can be expressed as: $(x, y, z) = x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1)$.
• 🌐 In $R^n$, the standard basis consists of $n$ vectors, where each vector has a single '1' in the $i$-th position and '0's elsewhere. For instance, the $i$-th basis vector $e_i$ is $(0, 0, ..., 1, ..., 0)$.
• 📜 For polynomials, the standard basis for the space of polynomials of degree $n$ or less, denoted $P_n$, is given by $\{1, x, x^2, ..., x^n\}$. Any polynomial $p(x)$ in $P_n$ can be written as $p(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$, where $a_i$ are coefficients.
• 💡 Understanding standard bases simplifies many linear algebra operations and provides a common reference point.

🧪 Practice Quiz

1. Question 1: What is the standard basis for $R^2$?
1. A) $\{(1, 1), (0, 0)\}$
2. B) $\{(1, 0), (1, 1)\}$
3. C) $\{(1, 0), (0, 1)\}$
4. D) $\{(0, 1), (1, 1)\}$
2. Question 2: Which of the following is a vector in the standard basis for $R^3$?
1. A) $(1, 1, 1)$
2. B) $(1, 0, 0)$
3. C) $(1, 1, 0)$
4. D) $(0, 1, 1)$
3. Question 3: How many vectors are in the standard basis for $R^5$?
1. A) 4
2. B) 5
3. C) 6
4. D) 7
4. Question 4: What is the standard basis for the space of polynomials of degree 2 or less, $P_2$?
1. A) $\{1, x\}$
2. B) $\{1, x^2\}$
3. C) $\{x, x^2\}$
4. D) $\{1, x, x^2\}$
5. Question 5: Which of the following vectors is NOT part of the standard basis for $R^4$?
1. A) $(1, 0, 0, 0)$
2. B) $(0, 1, 0, 0)$
3. C) $(0, 0, 1, 0)$
4. D) $(1, 1, 0, 0)$
6. Question 6: If $p(x) = 3 + 2x - x^2$, what are the coefficients with respect to the standard basis of $P_2$?
1. A) $(1, 2, 3)$
2. B) $(3, 2, -1)$
3. C) $(-1, 2, 3)$
4. D) $(2, -1, 3)$
7. Question 7: What is a key property of the vectors in a standard basis?
1. A) They are linearly dependent.
2. B) They span a subspace.
3. C) They are linearly independent.
4. D) They are non-zero vectors, but do not span the space.