Preserving a basis by replacing a vector

Dependencies:

  1. Basis of a vector space
  2. A set of dim(V) linearly independent vectors is a basis

Let $B = [u_1, u_2, \ldots, u_n]$ be a basis of a vector space $V$. Let $w = \sum_{i=1}^n \alpha_i u_i$. Then $B' = [u_1, u_2, \ldots, u_{k-1}, w, u_{k+1}, \ldots, u_n]$ is a basis of $V$ iff $\alpha_k \neq 0$.

Proof

Without loss of generality, assume $k = n$. Since $B'$ contains $n = \dim(V)$ vectors, $B'$ is a basis of $V$ iff $B'$ is linearly independent. We will now try to prove that $\alpha_n = 0$ iff $B'$ is linearly dependent.

Assume $\alpha_n = 0$. Then $b = \sum_{i=1}^{n-1} \alpha_i u_i$, so $B'$ is linearly dependent.

Let $B'$ be linearly dependent. Then for some non-zero vector $[\mu_1, \mu_2, \ldots, \mu_n]$, we have \begin{align} & \sum_{i=1}^{n-1} \mu_iu_i + \mu_nb_n = 0 \\ &\iff \sum_{i=1}^{n-1} \mu_iu_i + \mu_n\left(\sum_{i=1}^n \alpha_iu_i\right) = 0 \\ &\iff \sum_{i=1}^{n-1} (\mu_i + \mu_n\alpha_i)u_i + \mu_n\alpha_nu_n = 0 \end{align}

Since $B$ is a linearly independent, we get $\mu_n\alpha_n = 0$ and $\mu_i + \mu_n\alpha_i = 0$ for all $1 \le i < n$. If $\mu_n = 0$, then $\mu_i = -\mu_n\alpha_i = 0$, which is a contradiction since $[\mu_1, \mu_2, \ldots, \mu_n]$ is a non-zero vector. Hence, $\mu_n \neq 0$, and so $\alpha_n = 0$.

Dependency for: None

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Integral Domain
  8. Comparing coefficients of a polynomial with disjoint variables
  9. Field
  10. Vector Space
  11. Linear independence
  12. Span
  13. Incrementing a linearly independent set
  14. Semiring
  15. Matrix
  16. Stacking
  17. System of linear equations
  18. Product of stacked matrices
  19. Matrix multiplication is associative
  20. Reduced Row Echelon Form (RREF)
  21. Matrices over a field form a vector space
  22. Row space
  23. Elementary row operation
  24. Every elementary row operation has a unique inverse
  25. Row equivalence of matrices
  26. Row equivalent matrices have the same row space
  27. RREF is unique
  28. Identity matrix
  29. Inverse of a matrix
  30. Inverse of product
  31. Elementary row operation is matrix pre-multiplication
  32. Row equivalence matrix
  33. Equations with row equivalent matrices have the same solution set
  34. Basis of a vector space
  35. Linearly independent set is not bigger than a span
  36. Homogeneous linear equations with more variables than equations
  37. Rank of a homogenous system of linear equations
  38. Rank of a matrix
  39. A set of dim(V) linearly independent vectors is a basis